The Maxwell class exact solutions to the Schrödinger equation and continuum mechanics models

1 THE MAXWELL CLASS EXACT SOLUTIONS TO THE SCH RÖDINGER EQUATION AND CONTINUUM MECHANICS MODELS E.E. Perepelkin a,c,d,* , B.I. Sadovnikov a , N.G. Inozemtseva b , A.S. Medvedev a a Faculty of Physics, Lomonosov Mosc ow State University, Moscow, 119991 Russia b Moscow Technical University of Communi cations and Informatics, Moscow, 123423 Russia c Dubna State University, Moscow region, Dubna, 141980 Russia d Joint Institute for Nuclear Rese arch, Moscow region, Dubna, 141980 Russia ∗ Corresponding author: pevgeny@jinr.ru Abstract By applying the nonlinear Legendre transform to the continuity equation, this paper derives exact solutions to the Schrödinger equation and the equ ations of continuum mechanics. A generalized Maxwell distribution has been used as t he momentum density function. Explicit expressions for the vector fields of time independent flows, density distributions, quantum and classical potentials have been f ound, and a detailed mathematic al and physical analysis of t he results obtained has been carried out. Key words: exact solution, Schrödinger equation, Legendre transform, nonli near partial differential equation, continuum m echanics, g eneralized Maxwell distribution, rigorous result. Introduction The continuity equation is a f undamental equation of mathemati cal physics. It is used in continuum mechanics [ 1–3 ], astrophysics and gravity [ 4–9 ], electrodynamics [ 10–12 ], statistical physics [ 13–15 ], plasma physics [ 16–18 ] and quantum mechanics [ 19 ]. Note that the continuity equation is the first equation in Vlasov’s infinite self-linked chain of equations [ 20, 21 ]. Here ar e the first two equa tions of the chain () () () 11 ,d i v , , 0 , tr fr t fr t v r t ∂+  =     (i.1) () () () () 22 2 ,, d i v ,, d i v ,, ,, 0 , tr v fr v t fr v t v fr v tv r v t  ∂+ + =           (i.2) where () ( ) 3 def 3 12 ,, , , f rt f rv t d v =     () () ( ) 3 3 12 ,, , , , f rt v rt f r v t v d v =        (i.3) () () () 3 3 23 ,, ,, ,,, . f rv t v rv t f rv v t v d v =           Depending on the problem under consideration, the functions n f may have various physical interpretations. In hydrodynamic and gas-dynamic probl ems, n f corresponds to mass density; in electrodynamics to c harge density; in quantum mecha nics to the probability density of a single particle; and in statistical physics, plasma physics, astrophysics and high-energy physics to a distribution function or probability density. The index « n » indicates the dimension of the generalized phase space of higher kinematic quantities [ 22 ]. For example, when 1 n = t he function () 1 , f rt  is defined in coordinate space, when 2 n = the function () 2 ,, f rv t   i s considered in phase space, and when 3 n = the function () 3 ,,, f rv v t    is defined on an extended (generalized) phase space that takes the acceleration v   into account. It should be noted that the 2 power of electromagnetic radiation is related to the force of r adiative friction, which is proportional to v   . The quantity v   appears in the third Vlasov equation and in the Lorentz– Abraham–Dirac equation [ 23, 24 ]. Th e v ec t o r f i e l d () , vr t  corresponds to the flux velocity i n coordinate space. Its seco nd averaging over coordinate space with the function () 1 , f rt  yields () vt  – the velocity of the s y s t e m ’ s c e n t r e o f « m a s s » ( h e r e « m a s s » i s r e f f e r e d t o a s a g e n e ral meaning and in particular as probability). The field () ,, vr v t    corresponds to the accelerati on specified in phase space. Repeated averaging over the function () 2 ,, f rv t   yields the acceleration field () , vr t   at each coordinate point in the medium. Triple averaging () vt   leads to the acceleration of the system’s centre of «mass». Integrating the second Vlasov e quation (i.2) over the entire ve locity space transforms it into the first Vlasov equation (i.1). If we multiply equation ( i.2) by the velocity component k v and integrate it over the velocity space, we obtain the equatio n of motion for continuum mechanics () 1 1 , kt r k k k d vv v P v dt f λλ =∂ + ⋅ ∇ = − ∂ +   ( i . 4 ) () () 3 3 2 , kk k Pf v v v v d v λλ λ =− −   ( i . 5 ) where the second-order moment k P λ corresponds to the pressure tensor. Thus, the left-hand side of equation (i.4) contains the t otal time derivative of the medium’s flow velocity k v , whilst the right-hand side contains the for ce density, consisting of the p ressure force 1 k Pf λλ −∂ and the external force ~ k v  . Multiplying the second Vlasov equation by 2 v and subsequently integrating it over the velocity space yields the law of energy conservation 3 22 3 11 2 11 1 Tr Tr Tr , 22 2 2 2 t kk s s s kk k ks kks k k ff vP v v v P v P P f v v d v    ∂+ + ∂ + + + =           () () () 3 3 2 . kns k k n n s s Pv v v v v v f d v =− − −   ( i . 6 ) T h e f i rs t t e r m o n t h e l ef t -h an d s id e o f ( i . 6 ) c o r r e s p o n d s t o t he time derivative ( t ∂ ) of the energy density, which consists of the sum of the kinetic energy 2 1 2 fv  a n d t h e i n t e r n a l energy Tr 2 kk P . The repeated indices mean summation (the Einstien rule). The second term on the left-hand side of (i.6) corresponds to the divergence ( s ∂ ) of the energy flux density. The term 2 1 2 s fv v  is the kinetic energy flux density, whilst Tr 2 sk k vP is the internal e nergy flux density. The quantity kk s vP is related to the work done by gravitational forces, and Tr 2 kks P corresponds to the heat flux [ 20 ]. The integral on the right-hand side of equation (i.6) accoun ts for the power work of external forces. Another importan t property of eq uations (i.1)-(i.2 ) is their c onnection with quantum mechanics. Equation (i.1) reduces to the electromagnetic Schröd inger equation upon substituting 3 () 2 1 0, ex p fi ϕ =≥ = ∈ ψψ ψ  and using the Helmholtz decomposition into potential and vortex fields [ 25, 26 ] () () () ** * ,, , ln Arg Ln , r rr vr t r t A r t ii A i A αγ α γ α γ =− ∇ Φ + =  =∇ + + =∇ +    ψ ψψ ψψ ψ       ( i . 7 ) () def ,2 2 , , rt k k ϕπ Φ= + ∈   ( i . 8 ) where ,, α βγ are constants. Substitu ting (i.7)-(i.8) in to (i.1) yields the equations 2 ˆ p, 2 t i A U γ αβ βα β  ∂= − − +   ψ ψψ  ( i . 9 ) 2 11 VH , 2 t v ϕ βα β −∂= − + =  VQ , Q , r U α β Δ =+ = ψ ψ (i.10) () () , tr d vv E v B dt γ =∂ + ⋅ ∇ = − + ×     ( i . 1 1 ) def def 2 V, curl , tr r EA B A α β γ =−∂ − ∇ =    ( i . 1 2 ) 2 21 div V 0, rt A c α β γ +∂ =  ( i . 1 3 ) where def ˆ p r i β =∇ , () , Ur t ∈   . When the constants 2 m α =−  , 1 β =  , qm γ =− are chosen, equation (i.9) reduces to the electrom agnetic Schrödinger equat ion for the wave function ψ . Here,  is the Planck constant, m i s the mass, and q is the charge. The operator ˆ p takes on the role of the momentum operator, and the function U that of the potential energy. Equation (i.10) coincides with the Hamilton–Jacobi equation, i n which the energ y V is the sum of the potential energy U and the quantum potential Q . The quantum potential Q a p p e a r s i n t h e d e B r o g l i e - Bohm «pilot-wave» theory [ 27–29 ]. Equation (i.11) corresponds t o the equation of motion of a charged particle i n electromagnetic fields (i.12). The vortex f ield () , Art   from the Helmholtz decomposition corresponds to the vector potential of the magnet ic field B  . Condition (i.13) is the Lorenz Ψ -gauge and breaks down into t he standard Lorenz gauge in field theory and the gauge for the quantum potential [ 26 ]. From a comparison of the equations of motion (i.4) and (i.11) in the absence of magnetic fields it follows t hat the gradi ent of the quantum potential (f orce) can be interpreted as the quantum pressure force 1 Q. kk m P f λλ ∂= ∂ ( i . 1 4 ) We also note two important special cases of the second Vlasov equation (i.2). The first case is when there are no sources of dissipation 2 div 0 v Qv ==   [ 3 ]. In this case, equation (i.2) 4 reduces to the well-known Liouville equation. The second case i s when v   a d m i t s t h e V l a s o v - Moyal approximation [ 30 ] () () () () () () () () 12 2 2 2 21 0 12 ,, ,, , ,, . 21 ! nn n kk r v n n f rv t v rv t U rt f rv t mn + +∞ + = − =∂ ∇ ⋅ ∇ +         (i.15) Substituting (i.15) into the second Vlasov equation (i.2) tran sforms it into the Moyal equation [ 31 ] for the Wigner function [ 32–33 ] () ( ) 3 2 ,, , , f rv t m W r p t =    of a quantum s ystem in phase space. Note that averaging (i.15) over the velocity space yields an expression for the right- hand side of the equation of motion (i.4) () () 1 ,, . kk vr t U r t т =− ∂    ( i . 1 6 ) Thus, all quantum corrections (terms with coefficients 2 n  ) disappear. In the classical limit, when 1  , only the first term remains i n the approximation (i.15), that is, . kk mv U =− ∂  From the expressions (i.1)-(i.16) given above, a deep fundament al connection between classical and quantum physics follows within the framework of t he so-called Wigner-Vlasov formalism , based on the first two equ ations of the infinite sel f-linked Vlasov chain equations. Consequently, knowing the solution to equation (i.1) or (i.2), one can c onstruct solutions to the Schrödinger equation (i.9 ), the Ha milton-Jacobi equation ( i . 1 0 ) , t h e e q u a t i o n s o f m o t i o n (i.4), (i.11), find the quantum potential (i.10) and the pressu re force (i.14), construct the vector field of t he flow of a continuous medium () , vr t   or the probability flux (i.7), and determine the space distribution of matter () 1 , f rt  or the probability density. All the above mentioned makes it possible to obtain all th e inform ation regarding both classical and quantum systems. The existence of exact solutions to nonlinear problems in mathe m atical physics is of great importance for at least two reasons. The first is methodologica l. The theory of nonlinear partial differential equations is considerably more complex than that of linear equations [ 34–35 ]. It is impossible to i ntroduce the concept of a general solution and i t is only possible to have a limited number of particular solutions. Finding each particular s olutio n to a nonlinear equation sometimes requires particularly sophisticated mathematical meth ods [ 36–43 ]. Knowledge of the exact solution to a nonlinear system allows one t o analyse the nature of its behaviour without resorting to computationally int ensive calculations. The second a s p e c t i s t h e u s e o f e x a c t solutions in the design and optimization of real physical insta llations [ 44, 45 ]. The fa ct is th at th e numerical methods used in software packages are not «perfect». Obtaining a correct calculated value requires the selec tion of a finite d ifference scheme, its order of accuracy, proof of the convergence of the iterative pro cess, and proof of the stabilit y of the numerical method. Such theorems i n the field of computational mathematics have been pr oven for a sufficiently broad class of linear equations [ 46 ]. For nonlinear equations, this issue has been studied to a mu ch lesser extent. Therefore, when writing program code that numerically solves a nonlinear problem, the question of t he correctness of t he result obtained always arises. Moreover, every numerical method has a multitude of free parameters that must b e selected in some way. For example, the PIC (Particle In Cell) method is widely used in mo delling plasma physics, astrophysics, accelerato r physics and hydrodynam ics. In this co ntext, the alg orithm for distributing charge density or mass requires the selection of a n adaptive finite-element mesh and the type of its approximation. Next comes the solution of the field equations, followed by the transition from the Eulerian to the Lagrangian grid, and then t he solution of the equations of motion. The procedure is multi-stage. How can one be sure that the result obtained is c orrect? 5 What should it be compared with? Therefore, the availability of exact s olutions to a nonlinear problem allows one to test the c orrectness of the numerical alg orithm and evaluate its characteristic parameters. In some cases, it is possible to emb ed the exact solution of a nonlinear equation into the finite-dif ference scheme itself, thereby incr easing the order of accuracy of the numerical algorithm [ 47–49 ]. The ai m of thi s w ork is t o c ons truct exact solu tions to nonline ar problems in classical and quantum physics based on the Wigner-Vlasov formalism. The start ing point is the first Vlasov equation (i.1), in which the density function 1 f has an explicit dependence on the flux modulus vv =  and an implicit dependence on the coordinates () () 1 ,, , f xy F v xy   =    ( i . 1 7 ) where F is a function to be chosen. When discussing distribution funct ions in momentum (velocity) space, one cannot overlook the well-known class of g eneralized Maxwell distributions () 22 ,, , , , , exp , 22 n nn vn v n n n Nz z Fz σσ   =−          ( i . 1 8 ) where 0 n > , ,, vn σ  is a characteristic lengt h i n t h e v e l o c i t y s p a c e v , and , n N  i s t h e normalisation factor in coordinate space. In the special case w here 2 n ==  , the distribution (i.18) () 2,2 F z formally coincides with the we ll-known Maxwell distribution. T he generalized Maxwell distribution (i.18) ( 1 n > ) is used in various branches of physics. At 4 n = it transitions into the Drüwestein distribution [ 50 ] and describes the energy distribution of electrons in a weakly ionized gas under conditions where the e lectron interaction cross-section begins to depend on their velocity. These conditions arise, for example, in certain astrophysical systems [ 51 ] and plasma physics [ 52 ]. In addition, the generalized Maxwell distribution finds a ppl ication in statistical physics. In the case of general-form power-law H am iltonians and an arbitrary number of degrees of freedom , th e Tsallis distribution yields t he single-particle generalized Maxwell-Tsallis distribution [ 53 ]. With a specific choice of par ameters, distribution (i.18) reduces to the Weibull distribution [ 54 ], which is widely used in statistics [ 55 ], engineering [ 56 ] and medicine [ 57 ]. Note that relation (i.17) is not coincidental, it is used in f ield theory when solving the nonlinear problem o f magnetostatics [ 48, 58–60 ]. Substituting (i.17) into (i.1) yields a time independent nonlinear second-order partial differential equatio n () () { } div , , 0, rr r Fx y x y α  ∇Φ ∇Φ =  ( i . 1 9 ) where, in the Helmholtz decomposition (i.7), only the potential component is considered. Note that for non-smooth potentials Φ , the field r ∇Φ may be vortical, i.e. curl rr θ ∇Φ ≠  . In t his case, it can be assumed that ~ r A ∇Φ  . Such an example will be consid ered in this paper. The paper is structured as follo ws. In §1 using the nonlinear L egendre transform, equation (i.19) is reduced into a linear partia l differential e quation with variab le coefficients. The coordinates in the new space are velocities (momenta). Dependin g on the type of momentum domain, the equation is of elliptic, parabolic or hyperbolic ty pe. The characteristic equations are derived and their properties investigated. The canonical form of the equations in each domain is obtained. In §2 exact solutions are constructed in momentum spa ce. The method of characteristics and the search for a solution in factored form (the product of the radial and 6 angular parts) are considered. The radial part of t he solution is represented by the Kummer confluent hypergeometric function, whilst the angular part is a s uperposition of trigonometric functions or a linear angular dependence. The case where the Ku mmer function reduces to the generalized Laguerre polynomials i s analysed in detail. In §3 a solution to the original nonlinear equation (i.19) is derived by m eans of the inverse Legendre tra nsform from momentum space to coordinate s pace. For the factor ized solution in m omentum space, an explicit expression is obtained for the solution Φ (i.19), the quantum Q and classical U potentials (i.10), the vector velocity field v  ( i . 7 ) a n d t h e d e n s i t y 1 f in coordinate space. A deta iled physical analysis is carried out of the system’s dynamics, the direction of the flow s and the forces acting upon the m. For the linear angular part in momentum space, an exact solutio n to the Schrödinger equation in coordinate space with a vortex field of probability flux veloci ty has been found. This solution belongs to the class of solutions of the so-called Ψ - model of micro- and macro-systems [ 61 ]. Relationships between the characteristic parameters of the dist ribution (i.18) and the standard deviations have been derived from the Heisenberg uncertainty pr inciple. The main perspectives of the work are presented in the conclusion. Proofs of theorems , lem mas and interm ediate calculations are given in Appendices A and B. §1 Nonlinear Legendre transformation Consider a two () , n  parametric probabili ty density function , n f  satisfying the time independent first Vlasov equation (i.1) () () ,, ,, . nn f xy F v xy   =     ( 1 . 1 ) The integral of the function () , n Fz  over the space of its argument z takes the form () 0 , , ,, 1 2, vn n n N Fz d z nn σ +∞ +  =Γ        (1.2) where Γ is the gamma function. We shall consider the vector field of t he probability flux to be a potential () () ,, . r vx y x y α =− ∇ Φ  (1.3) It should be noted t hat in the case of a non-smooth potential Φ , the expansion (1.3) may correspond to a vortex field, for example, the Ψ -model [ 61 ]. Substituting expressions (1.1) and (1.3) into the first Vlasov e quation (i.1), we obtain (i.19) () () () 2 2 ,, , 12 1 0 , xn r x x n r x y x y y n r y y hh h αα α    + Φ ∇Φ Φ + ∇Φ Φ Φ Φ + + Φ ∇Φ Φ =      ( 1.4) () () () 2 2 def , , 22 , ,, . 2 n n n nn n vn Fz nz hz zF z z α α σ  ′  == −        ( 1 . 5 ) Equation (1.4) is a nonlinear partial differential equation. I n the special case where the average velocities v satisfy the condition 7 1 ,, ~~ 0 , 2 n r vn v n σ   ΔΦ     ( 1 . 6 ) i.e. the phase of the wave function is close to a harmonic func tion. In the general case, according to [ 62 ], the nonlinear equation (1.5) admits linearisation using the nonlinear Legendre transformation [ 63 ] () ( ) ,, , x yx y ω ξ η ξ η +Φ = + ,, , , xy xy ξ η ξ ηω ω =Φ =Φ = = (1.7) ,, , xx xy yy JJ J η η ξ η ξξ ωω ω Φ= Φ = − Φ = () 1 22 , xx yy xy J ξξ η η ξη ωω ω − =Φ Φ − Φ = − where J is the Jacobian of the Legendre transform ation. Applying trans form ation (1.7) to equation (1.5), we arrive at a linear equation with respect to the function ω () () () 22 ,, , 12 1 0 , nn n hh h ηη ξη ξξ ξρ ω ξ η ρ ω η ρ ω   +− + +=     (1.8) () () 2 def ,, 22 ,, , 2 n n nn nn vn n hv h αρ ρ ρσ − == −    where 22 2 ρξ η =+ , v αρ = . We determ ine the t ype of equation (1.8) by calculating its determinant 2 12 11 22 aa a Δ= − , where 12 11 22 ,, aa a are the coefficients of the eq uation () () () () () 22 2 2 2 2 ,, , , , 11 1 , nn n n n hh h ρξ η ρξ ρ η ρρ ρ    Δ= − + + = − −       () ,, , 2 1. 2 n n n n n vn n ρ αρ σ Δ= − −    ( 1 . 9 ) According to (1.9), the determ inant , n Δ  depends only on the polar radius ρ . Consequently, the regions of ellipticity, parabolicity and hype rbolicity of equation (1.8) have radial symmetry (see Fig. 1 , right) , , , ,0 , ,0 , , ,0 , , n Tn Tn T elliptical type p arabolic type hyperbolic type ρρ ρρ ρρ  <Δ <  =Δ =   >Δ >     , 1 ,, , . 1 22 TT vn n nv v n αρ σσ    = =  +   (1.10) Note that the expression for the quantity T ρ (1.10) is close to the expression (1.6). Thus, in t he vicinity of the parabolic region ~ T ρρ (see Fig. 1 , right), the phase function Φ o f t h e wave function is close to a harmonic function. Since the parame ter ,, vn σ  in the distribution (1.1) is free, without loss of generality, we define it as 1 def ,, ,, 1 2 vv TT v n nv n v σσ σσ ρ α  == = +       (1.11) 8 where v σ is a constant that determines the region (1.10) of ellipticity , parabolicity and hyperbolicity of equation (1.8). For example, when 2 n = and 0 =  , distribution (1.1) becomes a Gaussian distribution, and v σ is its standard deviation. If we assume that the quantity v σ f o r m a l l y s a t i s f i e s t h e Heisenberg uncertainty principle 2 vr σσ ≥  , then the region of ellipticity of equation (1.8) is bounded by 1 T r ρσ ≤ . The representation (1.11) is convenient in t hat the quantity T ρ does not explicitly depend on the parameters , n  . Figure 1 (left) shows the distribution graphs of (i.18) for the parameters 1, 2 n = and 0, 1 , 2 =  . On the horizontal axis of Figure 1 (left), the quantity v is plotted in units of v σ . The solid line in the graph in Fig. 1 (left) corresponds to 2 n = , whilst the dotted line corresponds to 1 n = . According to (1.11), each distribution , n F  h a s i t s own value ,, vn σ  , but T v is the same for all of them (see Fig. 1 , left). Equation (1.8) takes a compact form in polar coordinates: cos ξρ θ = , sin ηρ θ = , ()( ) ,, u ω ξ η ρ θ = () , 2 11 0, n ug u u ρρ ρ θθ ρ ρρ  ++ =    ( 1 . 1 2 ) () () () ( ) () def 2 , 2 ,, , , 11 1 n nn n n T n gh ρ ρρ ρ ρρ ρρ  =+ = − +  −  =Δ = − Δ      (1.13) where 2 ,, nn g ρ Δ= −  is the determinant of equation (1.12). Let us consider the que stion of reducing equation (1.12) to canon ical form . The following state ment holds. Theorem 1. If 1 >−  , then equation (1.12) has the characteristic () () , , 2 ,1 a r c t g 1 , , 1 h nn nn T n T T n χ ρ ρθ ρ ρ ρ ρ θρ ±  =− − ± >    +−   (1.14) () () , , 2 , 1 arcth , 0 , 11 nn nn T e T n T n ρρ ρ χ ρ ρθ θ ρ ρ ± +−  −  =− ± < <      (1.15) Fig. 1 Form of equation (1.14) depending on the region 9 which reduce it to canonical fo rm for the regions (1.10) of ellipticity and hyperbolicity, respectively () () () () () () () () () () () () ,, 0, 0, eh nn χ χχ χ χχ χ χ χχ κκ ++ −− + − +− + −   ΩΩ − Ω Ω = Ω Ω Ω =   ++ +    +  (1.16) () () () ( ) () 2 , de , , f , 32 12 2 4 , n e n n n nn κρ ++ − + ΔΔ −Δ =     () () () ( ) 2 def , , 32 , , 12 2 , 8 h n n n n nn κρ Δ ++ − Δ = Δ −      (1.17) where () () () () () ,, ,, ,, , , eh eh nn u ρθ ρθ ρ χχ θ +−  =Ω   . In the region of parabolicity, T ρρ = , the solution () , T u ρθ is an arbitrary smooth periodic func tion with respect to the argument θ , that is () ( ) ,, 2 TT uu ρθ ρθ π =+ . The proof of Theorem 1 is given in Appendix A . Note that the implicit dependence of the functional coefficient s (1.17) on the variables () , , eh n χ ±  leads to difficulties i n constructing exact solutions to the e quations (1.16). Nevertheless, in certain special cases (see §2) it is possible to construct a n exact solution to the hyperbolic equation (1.16) in explicit form. In this connection, let us ex amine in more detail the behaviour of the characteristic curves (1.14)-(1.15). Figure 2 shows the graphs of the characteristic curves (1.14)-(1.15) for 2 =  a n d 1, 2 , 3 n = in the three regions (1.10). The black line in Figure 2 indicates the boundary T ρρ = (a circle) on which equation (1.12) is parabolic. In Fig. 2 , the main focus i s on the region ( T ρρ > ) of hyperbolicity of equation (1.1 2). In the e lliptic region ( T ρρ < ), the behaviour of the characteristics does not depend significantly on the parameter n . Therefore, as an example, Fig. 4 shows the characteristics for 2 n ==  . The characteristics are plotted with a step size of 3 π along the angle () , , eh n χ ±  . The angle between the chara cteristics and the circle T ρρ = at their point of intersection is a straight lin e (see Fig. 4 ). Fig. 2 shows that the density of the characteristic lines in the radia l direction at T ρρ > (nodal points along the horizontal axis ξ ) depends significantly on the parameter n . This dependence is determined by the graphs of the function θ ρ ′ in the corresponding colours in Fig. 3 . The values of the function θ ρ ′ d e t e r m i n e t h e t a n g e n t o f t h e a n gle of inclination of the tange nt to the graph of the characteristic (1.1 4)-(1.15) Fig. 2 Density of characteristic lin es in the hyperbolicity region 10 () () , , n θ ρ ρρ ρ ′ = ±Δ  ,, ,, T T ρρ ρρ +>   −<  ,0 2 , lim 1 , 2, 0, 2, T n n n θ ρ ρρ →∞ +∞ < <   ′ =+ =   >   (1.18) where, for T ρρ = , the determinant () , 0 nT ρ Δ=  and the function θ ρ ′ tend to infinity, which corresponds to the angle of the tangent 2 π (see Fig. 4 ). Figure 3 shows three graphs of the function (1.18) with the parameters 1, 2 , 3 n = and 2 =  . All three curves have a pole at the point T ρρ = , which corresponds to the parabolic region of equation (1.12) and separates the elliptic region ( T ρρ < ) f r o m t h e hyperbolic region ( T ρρ > ). The parameters 1, 2 , 3 n = were not chosen at random. The fact is t hat, depending on the value of n , the behaviour of the tangent θ ρ ′ ( 1.18) varies. When 2 n = , the graph of the function (1.18) has a horizontal asymptote 10 T ρ +≠  (see Fig. 3 , blue line). Recall that the parameter 2 n = in distribution (i.18) corres ponds to the Gaussian distributio n (when 0 =  ) or the M axwell distribution (when 2 =  ). In the case of 2 n = (see Fig. 2 , blue curve), for large radii ρ the characteristic curves have an alm ost constant slo pe towards the circle, which leads to t he conservation of the distance between the characteristics (see Fig. 2 , nodal points). For 2 n > , the function θ ρ ′ h a s a horizontal asymptote at zero (1.18) (see Fig. 3 , red graph). As a result, the angle of inclination of the tangent to the characteristic curve tends to zero and the distance between t he c haracteristics asymptotically tends to zero (see Fig. 2 , red nodal points). When 2 n < , the function θ ρ ′ i s non-monotonic. In Fig. 3 (green graph), the minimum of the function θ ρ ′ is visible at the point min 12 n T n ρρ =− . In the interval min T ρρ ρ << , a decrease in the angle of the tangent’s slope is observed, and when min ρρ > , the angle of the slope begins to increase smoothly. Consequentl y, the distance between the characteristic curves increases indefinitely (see Fig. 2 , green nodal points). I n the region of ellipticity 0 T ρρ << , the graphs of the characterist ics (1.15) exhibit strong spira l curvature near the origin (see Fig. 4 ). Indeed, for 0 ρ → , the function (1.15) contains arct h 1 nn T ρρ −→ ∞ . Before proceeding to the direct derivation of exact solutions to e quation (1.12), let us consider another form of its expression based on the method of separation of variables () ( ) ( ) ,~ uR ρθ ρ θ Θ Fig. 3 Tangent of the angle of inclination of the tangent to the characteristic curve Fig. 4 Graphs of characteristics in the region ellipticity of equation (1.12) 11 () 2 2 , 2 1 0, 0, n Rg R R λ ρλ ρρ  ′′ ′ ′′ +− = Θ + Θ =    (1.19) where const λ = . The function Θ takes the form () 12 12 ,0 , sin cos , 0, сс сс λ θλ θ λθ λθ λ +=  Θ=  +≠  (1.20) where 12 , сс are certain constants. In the case of periodic boundary condit ions, () ( ) 2 λλ θπ θ Θ+ = Θ t a k e s t h e v a l u e k k λ =∈  . Note that satisfying th e periodicity conditions is not mandatory. The reason is that, when performing the inver se Legendre transform back into coordinate space, to conserve the uniqueness of the transform, it is necessary t o map only the angular sector 12 θθ θ ≤≤ . This issue will be discussed in m ore detail in §3. Let us consider equation (1.19) in more detail for the function R . Theorem 2 . Equation (1.19) for the function () R ρ reduces to the Hill equation for the function () () RR ς ρς =   () , 0, n RG R ςς ρ ′′ +=  ( 1 . 2 1 ) () () () () 2 1 , ,, 2 0 , ,1 , 1 1, ,1 , n n n n T n n n Ge c ρ ϑρρ λ ρϑ ρ ρ ρ ϑρ ρ ρ + −  ≥ +  == −  −<         (1.22) upon substitution () () () () , ,0 0 ,, 1 ,, ! nk k nk nT k k Jn k c nk nn k k ρ ς ρρ ρ − +∞ =  =  =  + ≠  −         ( 1 . 2 3 ) () () , ,0 def 1 , ,, ,1 11 ,1 ,1 1 Ei , 0 , ,, n nk n n kn x nk nk nk nk kn k n nk nk xk n Jx x e Jk kk n x β β ββ ββ − − −−  =   =    −∈       ( 1 . 2 4 ) () def 1 E i l n v.p. , , ! x kt e k xe xx d t x kk t γ +∞ = −∞ =+ + = ∈    ( 1 . 2 5 ) where T ρρ ρ = , , 1 nk kn β =+ and () Ei x is the integral exponentia l function with the Euler– Mascheroni constant e γ . The proof of Theorem 2 is given in Appendix A . Remark 1. For greater clarity when writing equation (1.21), the derivativ e s w i t h r e s p e c t t o t h e variable , n ς  are denoted by the derivatives with respect to the variable ς . Thus, for each pair of parameters , n  there is a corresponding variable substitution (1.23) and a fu nction , n G  . Note 12 that the function , n G  explicitly depends on the variable ρ , whereas for the analysis of equation (1.21) a dependence on the variable ς is required. The explicit form of expressions (1.22)-(1.25) a l l o w s u s t o c o n s t r u c t s u c h a d e p e n d e n c e . F i g . 5 (left) shows the distributions of , n G  with respect to , n ς  f o r 2 n = and 0 >  , () 0.6 0.3 s s + =+  , 0...4 s = , whilst Fig. 5 (right) shows the distributions for 2 n = and 1 0 −< <  , () 0.75 0.15 s s − =− +  . The value () , nT ςρ  corresponds to the point whe re the graph of the function , n G  intersects the ς a x i s ( s e e F i g . 5 ; f o r e a c h g r a p h , t h e p o i n t i s m a r k e d w i t h a d i f f e r e n t c o l o u r ) . A t the point () , nT ςρ  equation (1.12) is of the parabolic type , 0 n G =  . To the left of this point, at () , nT ςς ρ <  , equation (1.12) is of elliptic type and the value is , 0 n G <  , whilst to the r ight, at () , nT ςς ρ >  , equation (1.12) is hyperbolic and , 0 n G >  . The boundary between t he regions of ellipticity and parabolicity in Fig. 5 is marked by a vertical dotted line. From a physical point of view, equation (1.21) is similar to the oscillation equation fo r a system with variable frequency , n G  . In the hyperbolic region ( , 0 n G >  ), the frequency is a real function and is equal to , n ϑ  . Fig. 5 shows that near the point () , nT ςρ  the frequency , n ϑ  changes sharply: a sharp rise, followed by a smooth asymptotic decline to zero. This behaviour is characteristic of both systems with 0 >  ( se e Fi g. 5 , left) and systems with 1 0 −< <  ( see F ig. 5 , right). In the elliptic region , 0 n G <  , so when () , nT ςς ρ <  , the frequency is a purely imaginary function , n i ϑ ±  a n d the oscillations degenerate. Furthermore, for positive values o f the parameter 0 >  (see Fig. 5 , left), t he function , 0 n ϑ →  when , n ς →− ∞  ( 0 ρ → ). In this case, the solutions to equation (1.21) are close to linear functions, since 0 R ςς ′′ → . For negative values of the parameter 10 −< <  ( s e e F i g . 5 , right), the function , ~ n ϑρ →∞   a t 0 ρ → ( , n ς →− ∞  ) , w h i c h m a y formally correspond to solutions of equation (1.21) with a shar p decline or rise. We present another method for constructing a solution to the d ifferential equation (1.19) for the component () R ρ . From the theory of differential equations, it is known that t he solution to equation (1.19) for the function () R ρ can be represented as the produ ct of an analytic function and a singularity function at the pole 0 ρ = () () () 1 11 0 , k k k R с υ λ ρρ ρ +∞ = =  ( 1 . 2 6 ) Fig. 5 Graphs of the frequency coe fficient for equation (1.21) 13 where 1 υ is the root with the larg est real part of the characteristic e quation () () 22 ,, 01 0 0 . nn gg υυ λ  +− − =   ( 1 . 2 7 ) If 2 υ differs from 1 υ by a non-integer amount and 21 υυ ≠ , then the second linearly independent solution is given by () () () 2 22 0 , k k k R с υ λ ρρ ρ +∞ = =  ( 1 . 2 8 ) otherwise () () () () () 2 21 3 0 0 ln , k k k R с Rc υ λλ ρρ ρ ρ ρ +∞ = =+  (1.29) where 0 с is a constant, and () 1 k с , () 2 k с , () 3 k c are certain expansion coefficients to be determined, for example, by the Fubini method. It follows from expression (1.13 ) that () , 01 n g =+   , therefore, the roots of equation (1.27) () 22 1, 2 24 1 υλ =− ± + +   , () 22 12 41 . υυ λ −= + +  (1.30) From the condition 1 −<  it follows that the roots are 1, 2 υ ∈  . Depending on the values of  and λ , different variants of solutions (1.28) and (1.29) are possibl e . F o r e x a m p l e , f o r periodic conditions (1.20) k k λ =∈  and for 0 =  (Gaussian distribution) according to (1.30) 12 2 k υυ −= , which leads to solution (1.29). §2 Exact solutions in the momentum representation Let us proceed to the direct construction of exact solutions t o equation (1.8) in momentum space. W e begin with t he hyperbolic domain (see Fig. 1 ) , 0 n Δ>  (1.10). Hyperbolic type Theorem 3. In the hyperbolic domain , 0 n Δ >  , fo r 1 >−  , equation (1.16) has a partial solution of the form () () () () () ( ) () 1 ,2 def ,, ,, , 1 2 1 0 1, , 1 11 ,, ! , nk n kk n n k k hh n nn n Ic n k c e n cn k nk n k χχ ε μ ε + + − ∞ − −+ = − +  + Ω += + ++      =Ω =    +≠ −             (2.1) () () , ,0 def , ,, ,1 1 ,1 ,1 Ei , 0, ,, nk n k x nk nk nk nk kk nk nk xk Ix x e Ik kk x β β ββ ββ − − −−  =   =   −∈        (2.2) () () () () def ,, 12 12 ,, , 12 c 1 ar tg 2 hh nn n n n n μχ χ ε ε ++ − + −  =+ =     , (2.3) 14 where 12 , cc are constant values and 1 n n ερ =− . The proof of Theorem 3 is given in Appendix A . Remark 2. Note that equation (1.16) has a coefficient () , h n κ  , which depends im plicitly on the variables () , , h n χ ±  and explicitly on the radius ρ . Nevertheless, the specified dependence (1.17) allows us to find a partial solution (2.1) to equation (1.16). This result is linked to the transition from the canonical coordinates () , , h n χ ±  (1.14) to the coordinate () , n μ +  , which depends only on ρ . The transition from () , , h n χ ±  t o () , n μ ±  leads to a separation of the radial ρ a nd angular θ dependencies. Indeed, according to (1.14) and (2.3), the variab le () , n μ +  is expressed solely in terms of ρ , whilst () , n μθ − =  . Thus, the solution (2.1) obtained possesses angular symmetry and does not depend on the angle θ . F i g . 6 shows graphs of the function (2.1) in the coordinate axes () ( ) ( ) 1 ,, , cos nn n ς μμ +− =   , () ( ) ( ) 2 ,, , sin nn n ς μμ +− =   . The transition from the coordinate system () , , h n χ ±  to the system () , n μ ±  leads to the «collapse» of the regions of ellipticity and parabolicity ( see Fig. 1 , right) into a single point – the origin (see Fig. 6 ). On the left-hand side of Fig. 6 , the solution Ω corresponds to the distribution parameter (i.18) 2 n = , whilst on the right-hand side of Fig. 6 it corresponds to the parameter 4 n = . In Fig. 6 , for each value o f n , three graphs have been plotted, corresponding to different values of the parameter () 0.3 1 , 1...3 j jj =+ =  . To illustrate the relative positions of the surfaces, their graphs in Fig. 6 have been plotted only for the range () , 03 2 n μπ − ≤≤  . A comparison of the graphs on the left and right in Fig. 6 shows that an increase in the parameter n in the vicinity of the origin (f or small () , n μ +  ) leads to a gentler rise in Ω . Fig. 6 Graphs of partial solutions to the hyperbolic equation (1.16) 15 Parabolic type The region of parabolicity of equation (1.12) is a circle of r adius T ρ ( s e e F i g . 1 on the right), on which, according to (1.13), () , 0 nT g ρ =  . Consequently, () () , T uu ρ θθ = i s a n arbitrary smooth function, for example, the expression (1.20). Elliptic type Let us map the region of ellipticity onto the u nit circle using the change of variables employed in the previous theorems () () () def , T RR R ρρ ρ ρ == () () () ( ) () def ,, , 11 , n nn T n gg g ρρ ρ ρ ρ == = + −    (2.4) () 2 , 2 1 0, n Rg R R ρρ ρ λ ρ ρρ  ′′ ′ +− =    ( 2 . 5 ) where 01 ρ << . Knowing the explicit form of the coefficient () , n g ρ  , equation (2.5) can be reduced to a well-known form. Theorem 4. Equation (2.5) for the function () R ρ reduces to the Kummer equation for the function () T τ () () ,, ,, 0, nn Tb T a T ττ λ τ λ ττ ±±  ′′ ′ +− − =   (2.6) where () () 2 , ,, n a n λ λ υλ ± ± − =   , () () , ,, 2 n n b n λ λ υ ± ± ++ =    , () () ( ) 2 2 , 2, 1 2 λ υλ ± =− ± + +    (2.7) () () () () () , ,, ,, 1 ,, n nn RT n λ υ λλ ρρ ττ ρ ± ±± + ==    (2.8) whose solution () () ,, n T λ τ ±  can be expressed as a supe rposition of the Kummer function M and the Tricomi function Ψ () () () ( ) () () () () ,, 1 ,, ,, 2 ,, ,, ,, ,, . nn n n n Tc M a b c a b λλ λ λ λ ττ τ ±± ± ± ± =+ Ψ     (2.9 ) The proof of Theorem 4 is given in Appendix B . Remark 3. It f ollows from expression (2.7) that () , 0 λ υ + >  , and () , 0 λ υ − <  . The values of the numbers () , λ υ ±  coincide exactly with the previously given values of the roots 12 , υυ (1.30). Therefore, by default, we shall assume the equivalence of the n otations () 1, λ υυ + =  and () 2, λ υυ − =  . Even t he form of the solutions (1.26) and (1.28) is similar to the representation (2.8), with the sole difference that () ,, n R λ +  corresponds to the value () , λ υ −  and () ,, n R λ −  to the value () , λ υ +  . Indeed, the roots 1, 2 υ of the characteristic equation (1.27) for the Kummer equation (2.8) take th e form 16 () () () () () () 2 1 2 12 , , 2 12 , 2 2 , 4, " " , 4 1 0, 1 ~ , 1 1, 0 , ~ , " " . n n T n T n υ λ υ λ υυ λ τ τ υλ υ τ τ + −  == − +     =+ = −  ++ +       (2.10) The behaviour of the function () () ,, n T λ τ ±  in (2.10) corresponds to the properties of the Kummer and Tricomi functions. It is known that the Kummer funct ion M is unbounded at infinity (the case () ,, n T λ −  ), whilst the Tricomi function Ψ is bounded at infinity (the case () ,, n T λ +  ). Substituting the expressions (2.10) into the representation (2. 8) yields the singularity functions () () () , ,, ~ n R λ υ λ ρρ − +   (pole) and () () () , ,, ~ n R λ υ λ ρρ + −   (zero). The functions M and Ψ are also known as the Kummer functions of the first and second kind, and they are confluent hypergeometric functions. For exam ple, ()() 11 ,, ;; M ab z F ab z = , whilst Ψ can be expressed via M () () () () () () () 1 11 ,, ,, 1 , 2 , . 1 b bb ab z M ab z z M a b b z ab a − Γ− Γ − Ψ= + + − − Γ+ − Γ (2.11) Depending on the values of the parameters () ,, n a λ ±  and () ,, n b λ ±  , the Kummer f unctions transform into a wide range of well-known orthogonal polynomial s and special functions. As an example, consider the generalized Laguerre polyn omials () () k Lz α (see Appendix B ). Lemma 1. Let 1 >−  and 0 n > , 0 k ∈  and 1 λ ≥ then for the solution () () () ,, ,, ,, nn M ab z λλ ++  of equation (2.6) to map onto the ge neralized Laguerre polynomials () () k Lz α it is necessary that the parameters () , n  of the distribution (i.18) satisfy the c onditions () () () () () () 22 , 0 2 , 2 , , ,1 , , 0, 1, 1, k ak kn kn Mk z с Lz b kn α λ λ λλ α α α λλ + +   =− =−  −+ =      ≠ =+      − ≤−    (2.12) where () () () def 0 11 1 ck k αα =Γ + Γ + Γ + + and { } 0 0 =∪  . The proof of Lemma 1 is given in Appendix B . Using equations ( 2.12), one can determine the parameters () , n  of the distribution (i.18) to which the solution of equation (2.6) in the form of generali zed Laguerre polynomials corresponds. For example, for th e distribution (i.18) with para met er s 2 n ==  , the following generalized Laguerre polynomials and their corresponding number s are obtained λ 1 λ = , () () 2 0 Lz ; 5 λ = , () () 4 1 Lz ; 22 λ = , () () 5 2 Lz ; (2.13) 4 λ = , () () 7 5 Lz ; 21 λ = , () () 8 7 Lz ; 33 λ = , () () 10 12 Lz . In the case where 2 n = and the values of λ ∈  from expressions (2.12) imply that 17 2: λ = () () {} 2 1 0, Lz =  ; 3: λ = () () {} () () {} () () {} 17 7 3 12 3 ,2 0 ; ,4 ; ,0 ; Lz L z L z == =   (2.14) 4: λ = () () {} () () {} () () {} () () {} () () {} () () {} () () {} 59 28 17 11 12 3 4 74 1 1 7 56 7 90, ; 32, ; 14 , ; 6 , ; 2, ; 0, ; 6 7 1 , . Lz Lz L z L z Lz Lz L z == = = = = =− >−     The number of possible variants of (2.14) for each value of λ is determined by the inequality (2.12) over the range of values k . From the condition 0 α ≠ it follows that the Laguerre polynomials () () () 0 kk Lz L z = a t 1 >−  for the distribution (i.18) will be absent from the solution to equation (2.6). Thus, for the case (2.12), the solution to equation (1.12) is c onstructed from a superposition of functions of the form () ( ) () () () () ( ) () , , 1 ,, ,, ,, 1 ,1 . n n b kn n kn n TT uL n λ υ λ λ λ λ ρρ ρθ θ ρρ + + −    + =− Θ             (2.15) Note that the solutions (2.15) hold not only in the elliptic r egion of equation (1.12), but also in the parabolic and hyperbolic regions. Fig. 7 shows the graphs of the function (2.15) for 2 n = and 3 λ = . The values  and k are related by the relations (2.12) and were found earlier (2.14). Fig. 7 Graphs of the solutions (2.15) to equation (1.12) for and 18 The lower part of Fig. 7 corresponds to three isometric projections of the graphs of the function (2.15). The upper part of Fig. 7 shows three top views illustrating the contour lines of the function (2.15). Red corresponds to the maximum values, and b l u e t o t h e m i n i m u m v a l u e s . Since the generalized Laguerre pol yn omials grow substantia lly ( in magnitude) for large values of the argument, for clarity in Fig. 7 the values of the functions (2.15) are multiplied by the weighting function 22 e α ρ ρ − . This weighting function coincides with the weighting function o f the generalized Laguerre polynomials. In the region of elliptic ity ( T ρρ < ), the generalized Laguerre polynomials are monotonic functions. The appearance of radial oscillations (see Fig. 7 ) occurs at values of T ρρ > . The number of such maxima and minima is determined by the ord er k of the polynomial () k L α . Thus, in Fig. 7 on the left, the re is only o ne ra dia l ma xi mum/ mini mu m for () 17 1 L , whilst in Fig. 7 on the right t here are three for () 3 3 L . The azimuthal oscillations in Fig. 7 are determined by function (1 .20), which in this case is () 3 sin 3 θθ Θ= . Thus, by specifying the parameters () , n  o f distribution (1.1), one can obtain various explicit forms of solutions to equation (1.12) in the momentum space () , mm ξ η . The original nonlinear equation (1.4) for the phase of the wave function 2 ϕ =Φ is defined in coordinate space () , x y . Consequently, it is necessary to construct the inverse Legend re transform (1.7) of the solutions to equation (1.12) in momentum space to the solut ions to equation (1.4) in coordinate space. §3 Solution in coordina te representation The original first Vlasov equatio n (i.19) is defined in coordin ate space. In the preceding sections, the corresponding solutions to the first Vlasov equat ion in momentum space have been constructed. Thus, it is necessary t o perform the inverse Legen dr e transform (1.7) from momentum space () , mm α ξ η − to coordinate space () , x y . Due to the nonlinear nature of the Legendre transform, the soluti ons of the linear equation with v ariable coefficients (1.8) will become particular solutions of the original nonlinear partial d ifferential equation (i.19). As an example, let us consider the factorised solution () ( ) ~ uR ρ θ Θ of equation (1.8), described in detail in the previous paragraph. Theorem 5. Let () ,, , n u λ ρ θ  be a factorised solution of equati on (1.12), restricted to the origin, such that 0 λ ≠ and 1 λ ≠ then the corresponding particular solution ,, n λ Φ  of the nonlinear equation (1.4) takes the form () ( ) ( ) ,, ,, ,, ,, 1 , nnn xy u λλλ ρθ ρ  Φ= −    () () () () ,, ,, , , n n x u y λ λ λ ρ ρθ θ ρ θ   =    ϒ       (3.1) () () () () () () def def ,, , ,, ,, cos sin 1 1, , , , sin cos nn nn n nn TT ab n λλ λ λ θθ ρρ ρυ θ θθ ρρ ++ + −   + =+ + =            (3.2) () () () () def 1, 1, ,, l n ,, , ,, Ma b da ab M ab db M a b τ ττ ττ ++ ==  () def 12 12 tg ln , tg cc d dc c λλ λθ θλ θλ θ − ϒ= Θ = + (3.3) where 12 , cc are constant values from the repr esentation (1.20) of the function λ Θ . The Jacobian ,, n J λ  of the Legendre transforma tion (1.7) takes the form 19 () () () () ( ) () () ( ) { 2 ,, 12 2 2 2 ,, ,, , ,, , 4 , ,2 n nn n n n u Jg g λ λλ λ λ λ ρθ ρθ ρ ρ θ ρ λ ρ θ ρ −    =− + ϒ − + ϒ +          () ( ) } 42 , , n g λ λ ρ θ ++ ϒ  ( 3 . 4 ) () 1 ,, ,0 , nT e J λ ρθ − =  (3.5 ) where the angle e θ is determined by the condition () 0 e λ θ ′ Θ= . If 1 λ = , then () 1 ,, 1 ,0 n J ρθ − =  and the inverse Legendre transform is not possible. The proof of Theorem 5 is given in Appendix B . Remark 4. The result of Theorem 5 extends to the case of the second linea rly independent solution (2.9), expressed in terms of the Tricomi function () ,, ab τ Ψ . In this case, the function () ,, ab τ  t a k e s t h e f o r m () () ,, ,, ab ab τ ττ ′ ΨΨ . Since the function () ,, ab τ Ψ , according to (2.11), can be explicitly expressed in terms of the Kummer func tion () ,, M ab τ , the expression (3.3) for the updated function () ,, ab τ  is known. The angle e θ corresponds to an extremum o f the function λ Θ and can be expressed as () 0 2 e k θπ π θ λ =+ − , k ∈  , where 02 1 tg cc θ = . The point () , Te ρ θ lies in the parabolic region, in which the solution has only angular dependence (see §2). The Legendre transformation is a tangent transformation, thus (3.5) for () 0 e λ θ ′ Θ= is natural. In the case where 0 λ = , the results (3.1)-(3.3) of Theorem 5 remain valid for () ( ) 01 1 2 сс с θθ ϒ= + . For 1 λ = the solution is () ,, ,~ n uA B C λ ρθ ξ η ++  , i.e. it is the equation of a plane for which t he uniqueness o f t he Legendre tangent transformation is lost. Note that the function () ,, , n u λ ρ θ  in the condition of Theorem 5 does not, in the general case, coincide with the function (2.15). The expression (2.15) is one of the possible forms of the function () ,, , n u λ ρ θ  . When considering the boundary value problem for equation (1.12) i n a circular region of radius 0 ρ , one can formulate the Dirichlet problem with the boundary condition () ( ) 00 , uu ρ θθ = . In this case, the function () 0 u θ will yield the Fourier coefficients () () 12 , kk сс for the function (1.20) () λ θ Θ , where k k λ =∈  . The solution () , , n u ρ θ  to equation (1.12) will be expressed as a superposition of harmonics () ,, , n u λ ρ θ  and the inverse Legendre transform can be applied to it. In the inverse Legendre transform, the one-to-one corresponden ce may be violated in the form of a multivalent effect of the function. For example, a ci rcular region in momentum space 02 θπ ≤< m a p s t o a r e g i o n w i t h p o l a r a n g l e 0 0 φφ ≤< , where 0 2 φ π > . In su ch cases, in momentum coordinates, one may con sider a region in the form of an angular sector. As an example, we shall construct solutions i n coordinate spac e for cases (2.14)-(2.15). We shall begin by mapping the solution from the momentum domain o f ellipticity ( T ρρ < ) to the coordinate domain. Fig. 8 shows the distributions of the flow velocity () , vx y  (top three figures) and the density () , , n f xy  ( bottom three figures) for different values of λ a nd  . The parameter 2 n = is the same for all distributions. 20 For 2 λ = , 0 =  , the initial momentum region was a circle of radius T ρ , which was mapped onto the coordinate region in the form of a «diamond» (s ee Fig. 8 , top left and bottom left plots). In the case of 3 λ = , 4 =  , the momentum region was a semicircle 0 θπ ≤≤ , and the coordinate region was close to the shape of a «circle» ( see Fig . 8 , middle top and middle bottom graphs) with a vertical c ut at 2 φπ = (see Fig. 8 , white dotted line). For the values 4 λ = , 2 =  , the momentum region is a circular sector of two types. The firs t type has an angle of 02 θπ ≤≤ and the second 12 102 θ −≤ ≤  . The momentum region of the first type is m apped onto a circul ar sector with an angle of 2 πφ π −≤ ≤ . In Fig. 8 (top right) the vertic al and horizontal white dotted lines indicate the boundaries of the first-type region. The momentum region of the second type is mapped onto the coordinate space in the form of a «hear t» or a «leaf» of a tree with a n oblique cut along the black solid line (see Fig. 8 , top right). The red colour in Fig. 8 corresponds to the maximum values of the velocity distributio n modulus () , vx y and density () 2, , fx y  , whilst the blue colour corresponds to the minimum values. The direction of the velocity vector field () , vx y  in Fig. 8 (top three plots) is shown as small coloured dashes, as well as large white and black arrows. Let us consider each of the three cases 2, 3, 4 λ = separately. We begin with the vector field () , vx y  w h e r e 2 λ = a n d 0 =  ( s e e F i g . 8 , t o p l e f t ) . A f l o w c o m e s f r o m t w o d i r e c t i o n s (top left and bottom right) (black arrows) and then goes out in two directions (top right and bottom left) (black arrows). Inside this region the diagonal in coming flows collide and after scattering come out in diagonal directions. It can be seen that in the vicinity of the centre of t he flow collision, t he velocity is close to zero (blue), whilst th e maximum flow velocity is at the boundary of the region (red). At the corners of the region, white arrows indicate strong flow vortices v  caused by the proximity of the incoming and outgoing flows. Kn owing the Fig. 8 Graphs of the velocity and density in a coordinate syst em for the elliptica l case 21 distribution () , vx y one can find the correspond ing density distribution () , , n f xy  using formula (1.1), shown at the bottom of Fig. 8 . Since 0 =  , the distribution 2,0 f is the Gaussian distribution with a maximum at zero velocities (see Fig. 1 , left). Indeed, in Fig. 8 (bottom left), the maximum density is at the centre, where there i s almost no motion due to the collision of the flows. In the case of 3 λ = a n d 4 =  ( s e e F i g . 8 , centre), the flow enters from below and exits from above on the right and left sides ( black arrows) of the vertical section (white dotted line). The large white arrows show the characteristic directions of fl ow within the region, whilst the small coloured lines illustrate the flow in detail. The length of the white arrows corresponds to the magnitude of the flow velocity. Note that the vertical sect ion (white dotted line) i s the source of flow in t he horizontal directions (to the right and left). T he magnitude of the flow from the section increases with distance from the centre (the length of the horizontal white arrows increases). As in the previous case, the minimum velocity is in the central r egion. Since 4 =  , the maximum of the density function 2,4 f , unlike in the previous case, i s attained a t non-zero velocities. Indeed, in Fig. 8 (centre and bottom), large values (red) of the density 2,4 f ar e l o ca te d at the periphery of the coordinate domain, whilst in the centra l r e g i o n t h e d e n s i t y i s m i n i m a l (blue). As noted above, for the values 4 λ = , 2 =  , two types of c oordinate domains were obtained. The domain of the fir st type is a special case of the domain of the second type. A notable feature, which is the reason for its s eparate consideration, is that the direction of the flow v  at its boundary (white dotted line) is normal (orthogonal). In F i g . 8 on the right, the white arrows indicate norm al inflow through the vertical boundary and normal outflow through the horizontal boundary. As the angle of the solution domain increa ses, the problem of m ulti- leafness of the Legendre transfor mation for the coordinate doma i n m a y a r i s e . I n t h i s c a s e , a second-type domain with a wedge-s haped section is selected as a n extended continuation of the first-type domain. As can be seen in Fig. 8 (right), the flow direction v  continues into the extended coordinate part of the second-type domain. There is fl ow at the boundary of the wedge- shaped section. The flow enters through the lower part of the s ection boundary and exits through the upper part of the section boundary. The flow velocity is mi nim al in the central region (blue and green) and increases with distance from it (orange and red) . The length of the white arrows also indicates this. When 2 n = a n d 2 =  , the density distr ibution (i.18) is effectiv ely the Maxwell distribution (see Fig. 1 , left) therefore, as in the previous case ( 3 λ = ) , t h e d e ns i t y 2,2 f (see Fig. 8 , bottom right) i s minim al at the centre of the region (minim um velocities) an d increases in a certain nei ghbourhood (maximum velocity). Let us note one common feature of t he distributions shown in Fi g. 8 . All velocity distributions () , vx y  contain a coordinate region in w h i c h t h e v e l o c i t y i s z e r o . T h i s characteristic of the distributions () , vx y  is determined by the shape of the initial momentum domain. Indeed, all the momentum domains shown were either a ci rcle or a sector centred at the origin of the momentum space () ( ) ,0 , 0 mm αξ η −= , i.e. with zero velocity. The shape of t he momentum region influences the presence of different velocity d irections. For example, in the first case for 2 λ = and 0 =  , the momentum region was a circle, i.e. it contained all possi ble velocity directions v  ranging from 0 t o 2 π . Looking at Fig. 8 (top left), it can be seen that the distribution () , vx y  contains all directions from 0 t o 2 π . In the second case, for 3 λ = and 4 =  the impulse r egion is a semicircle with velocity directions v  from 0 and π . Consequently, in Fig. 8 (centre, top), the flow moves from bottom to top and diverges t o the right and left, corresponding to angles from 0 to π . A similar situation applies to the final case 4 λ = 22 , 2 =  . I n t h e f i r s t t y p e o f m o m e n t u m region, the direction of the ve locities v  ranges from 0 to 2 π , which is fully consistent with the distribution () , vx y  in Fig. 8 (top right). The sa me situation applies to the sec ond type of m omentum region. The second general feature of the distributions in Fig. 8 is the boundedness of the velocity magnitudes T v αρ < , caused by the size of the elliptic mom entum domain ( T ρρ < ) of equation (1.8). A greater variation in the magnitude of the velocity v can be obtained by mapping the hyperbolic momentum domain with T ρρ > . Thus, the variety of directions of the velocity vector fields v  is determined by the shape of the momentum domain, whilst the range of velocity magnitudes v is determined by its dimensions. Figure 9 shows the distributions of velocities v  and densities 2, f  f o r t h e m o m e n t u m region in which equation (1.8) is of the hyperbolic type. The general structure of Fig. 9 and the notation used therein are analogous to the structure and notation introduced in Fig. 8 . The same distribution parameters () , n  and values λ are used. The momentum region is a radial sector () 12 m a x ,: , , T ρθ ρ ρ ρ ρ θ θ <≤ ≤ ≤ (3.6) where the values 12 , ρρ a nd max θ are chosen from the condition of univalent of the coordinate domain in the inverse Legendre transform. From the form of the domain (3.6), it follows that the spread in the velocity directions v  ranges from max θ − to max θ and in the magnitude v ranges Fig. 9 Graphs of the velocity and density for the hyperbolic case 23 from 1 α ρ t o 2 α ρ . The following parameters of the momentum domains were used in constructing the distributions in Fig. 9 12 m a x 2, 0 : 1.8 , 2.4 , 12 , TT λρ ρ ρ ρ θ == = = =   (3.7) 12 m a x 3 , 4 : 1.5 , 1.89 , 15 , TT λρ ρ ρ ρ θ == = = =   (3.8) 12 m a x 4, 2 : 1.45 , 1.75 , 12 . TT λρ ρ ρ ρ θ == = = =   (3.9) In all three upper graphs in Fig. 9 , t he velocity distributions are consistent with the form of the momentum domain (3.6), i.e. 0 x v > , whilst y v may be positive, negative or zero. It follows from expressions (3.7)-(3.9) that when 2, 3, 4 λ = t h e m a x i m u m 2 α ρ a nd m inimum 1 α ρ values of the velocity decrease (the length of the white arrow s in Fig. 9 ). This behaviour of the velocity v also determines the nature of the density distribution 2, f  (the bottom three graphs in Fig. 9 ). Since the region of hyperbolic ity is under consideration (se e Fig. 1 , left), the velocity values lie at the tail (tail) of the density distributions 2, f  . Consequently, the minimum velocity value v leads to the maximum possible density value 2, f  , whilst the maximum velocity v yields the minimum density 2, f  . Consequently, in Fig. 9 (density plots 2, f  ), the upper part of the region has a red hue, whilst the lower part is blue. Since the minimum value v decreases as 2, 3 , 4 λ = , the size of the region with a red hue increases (see Fig. 9 ). When considering the hyperbol icity region of equation (1.8), r ecall the solution () , n μ +   Ω    (2.1), which corresponds to the case 0 λ = . Let us construct the inve rse Legendre transform for it. Lemma 2 . The solution () , n μ +  Ω   (2.1) of equation (1.16) for , 0 n Δ>  and 1 >−  in momentum space corresponds to the solution () , , n x y Φ  of the nonlinear equation (1.4) in coordinate space of the form () () () ,, , ,, nn n xy R ρς ρ ρ Φ= −   () , , n x y ξ ςρ η ρ    =        (3.10) () 1 ,0 1 exp , n T n n T c n ρ ρ ςρ ρρ +   + =        () () , , , n n R μ ρ +  Ω=    (3.11) where () , n R ρ  is the solution to equation (1.19) for 0 λ = , 0 с is a constant and in accordance with (2.3), it is taken into account that () () () ,, nn n μμ ε ++ =  and () nn εε ρ = . The proof of Lemma 2 is given in Appendix B . Remark 5. It follows from transformation (3.10) that () , n r ςρ =  and φ θ = , where cos xr φ = , sin yr φ = . Thus, there is an explicit rel ationship between the coordinat e radius-vector r  and the velocity () , vx y r α =  . Since there exists a minimum value min T ρρ = for the region of momentum hyperbolicity, there also exists a minimum radius (3.1 1) () ( ) min , exp 1 nT rn ςρ == +     in coordinate space, where 0 1 c = . 24 As an example, Fig. 1 0 ( left) shows the velocity vector field () , vx y  for the solution (3.10)- (3.11), whilst Fig. 1 0 (right) shows the corresponding density function () 2,0 , f xy f o r 2 n = and 0 =  . In constructing Fig. 1 0 , an momentum domain in the for m of a closed ring ( 02 θπ ≤< ) with an inner radius of T ρ and an outer radius of 2.5 T ρ wa s used, which corresponds to the hyperbolic type of equation (1.16). As mentioned previously, wh en mapping the momentum domain onto the coordinate domai n, there exists an «empty» regi on in the form of a circle of radius min r ( s e e F i g . 1 0 , right). The flux is directed along the radial lines and incre ases with the radius (Fig. 10, left). According to Fig. 1 (left), the hyperbolic region lies at the «tail» of the distribution 2,0 f , so the maximum density is at tained in the neighbourhood of th e mi nimu m radius min r (see Fig. 10 , right) . Theorem 6. Let () ,, , n u λ ρθ  be the factorised solu tion to equation (1.12), 0 λ ≠ , 1 λ ≠ and 1 >−  then the expression for the quantum potentia l () () Q, Q , r φρ θ = (i.10) takes the form () () () () () ( ) 2 2 ,, ,, ,, ,, , 2 ,, Q, 1 , , , , 2, nn n n n n g u λλ λ λ λ λ αρ ρθ ρ ρ λ ρ θ βρ θ  =− − ϒ         (3.11) () () () def , , 1234 , , 1234 , 1 234 ,, , ,, , ,,, , nn n zz z z zz z z zz z z λλ =+     () () () () 2 def 2 3 ,, 1 2 3 4 ,, 1 2 3 4 3 22 2 0 14 3 2 1 ,, , ,, , k k nn k z zz z z A zz z z zz z z λλ = − = +     (3.12) () () () () ( ) 22 2 def 2 14 2 , 1234 3 3 2 22 2 14 3 2 1 ,, , 1 1 1 , 2 n zz z zz z z z z n n zz z z +  =− + − − −   +     (3.13) () () () () ( ) {} () () () ( ) {} () 32 21 3 3 2 3 def 22 2 2 ,, 1 2 3 1 2 1 2 3 3 3 2 3 1 3 11 2 3 11 1 , 0 , ,, 2 3 1 3 1 , 1 , 1, 2 , k n zz z z z n z n k Az z z z z z z z n z n z z z z k zz z z k λ λ λ   −+ − + + =     =+ + − + + − − =      −− =      (3.14) where the relationship be tween the coordinate () , xy and momentum () , ρθ representations is determined by the transformation (3.1). The proof of Theorem 6 is given in Appendix B . Since the problem under consideration is time independent, the potential U can be found from the Hamilton–Jacobi equation (i.10). Indeed, the phase ϕ of the wave function ψ takes the form Fig. 10 Solution of (2.1) in the coordinate representation (3.10) 25 () () ,, E , rt r t ϕϕ φ β =−  ( 3 . 1 5 ) where the constant E corresponds to the energy of the system and the phase () () 2, , rr ϕφ φ =Φ is the solution to the original nonlinear equation (1.4). Consequently, equation (i.10) takes the form () () () 2 ,, ,, 1 ,, Q , E , 4 nn Ur v r r λλ φφ φ αβ =− +   (3.16) where the quantum potential ,, Q n λ  is given by expressions (3.11)-(3.14) and the kinetic energy is determined via the flux velocity v α ρ = . Remark 6. When calculating the quantum potential Q , an expression of the fo rm () ( ) QQ rr С =+   is obtained, where C const = and () Q r   is some function without a free constant. From a physical point o f view, in the eigen coordinat e system, the value is E C = . In this case, the potential (3.16) does not depend on an arbitrary constant. This situation is standard for quantum systems. A quantum system with potential U may have a set of wave functions s ψ corresponding to various energy levels E s . Each wave function s ψ corresponds to its own quantum potential () () 2 QQ E . 2 rs s s s rr m ΔΨ =− = + Ψ     ( 3 . 1 7 ) When substituting (3.17) into equation (3.16) the constant E s cancels out and the potential U is the same f or all quantum states with num ber . s Fig. 11 shows the distributions of the potential energy ,, n U λ  ( 3.16). For the sake of comparative analysis, the values of ,, n λ  correspond to those considered earlier in Fig. 8 . The top three graphs in Fig. 11 show the isometric projection of the potential function ,, n U λ  whilst the corresponding bottom three gr aphs are contour lines (top vi ew) of the potential function. Since the potential ,, n U λ  takes on large negative and positive values in the central reg ion, the values ,, n U λ  have been clipped to improve image contrast. Consequently, in Fig. 11 (bottom right and bottom centre), there are white areas where t he value s 2,4,3 U and 2,2,4 U exceed the set limits. Let us consider the physical int erpretation of the graphs in Fig. 1 1 . We shall begin with the cases 0 =  a nd 2 λ = (the top-left and bottom graphs in Fig. 1 1 ). As can be seen in Fig. 11 (top-left), the potential energy surface 2,0,2 U i s flat in the central region, but drops sharply as one moves away from it. Consequently, the external force r F U =− ∇  will be small in the central region and large at its boundary. Since the potential energy de c r e a s e s s h a r p l y a l o n g t h e r a d i u s , the force is directed away from the centre and is denoted in Fi g. 11 (bottom) by the vector F +  . This repulsive force counteracts the external, inflowing flows v  ( s e e F i g . 8 , top left). Upon encountering a sharp potential barrier, the flows v  reduce their velocity and enter the central region almost at rest (see the blue central region in Fig. 8, top left). As a result, the density in the 26 central region increases 2,0 f (the red central region in Fig. 8, bottom left). The accumulated flow in the centr al region is pus hed outwards by the force F +  ( i n F i g . 11, bottom left) along two diagonal directions (black arrows in Fig. 8, top left). For 4 =  and 3 λ = (the central top a nd bottom graphs in Fig. 1 1 ), the potential energy surface 2,4,3 U has a complex shape. In the central region, there are three «p etals» with an infinite potential barrie r and three «petals» with an infinitely deep well. These potential barriers and wells in Fig. 11 (bottom centre) correspond to the white regions in the centre. In the vicinity of the potential barriers, there is a repulsive force F +  , and in t he vicinity of t he potential wells there is an attractive force F −  ( s e e F i g . 1 1 , bottom centre). As shown in Fig. 8 (top centre), there is an incoming fl ow from below v  , which encounters a potential barrier in the for m of a lower vertical lobe (see Fig. 1 1 , bottom centre). As a result, a repulsive force F +  acts on the incoming flow v  , slowing it down. The potential ba rrier itself, in the form of a thin lobe, resembles a «knife» cutting the incoming flow in two (see Fig. 8 , centre top). Each of the two resulting flows i s a t t r a c t e d t o w ar d s t h e c e n tr a l r e g i o n by t h e f or c e F −  from the potential well (the blue region in Fig. 1 1 , centre bottom). Next comes anot her lobe of the potential barr ier, which impedes rectilinear motion and directs the flows in diagonal directions t owards the periphery of the region. A similar situation occurs with horizontal flows emergi ng from the vertical slit (see Fig. 8 , top centre). Each hor izontal flow formed in such a way co mes to the potential barrier peta l and exits the region in the corresponding diagonal direction. Accor ding to the density distribution 2,4 f (see Fig. 8 , bottom centre), the main flow v  flows along the perimeter of the region and i s scarcely present in its central part. Fig. 11 Potential energy (3.16) and char acteristic direction of externa l forces 27 In the latter case, for 2 =  and 4 λ = ( s e e F i g . 11, top right and bottom), the potential energy surface 2,2,4 U has a complex shape, partly resembling the previous distributi on 2,4,3 U . In the region under consideration 2,2,4 U has two potential barriers and three potential wells in the shape of petals (see Fig. 11, bottom right). This potential e nergy configura tion creates a characteristic distribution of forces F ±  shown in Fig. 11 (bottom right). According to Fig. 8 (top right, black arrow), the flow v  enters the region from the bottom left diagonally. Next, the incoming flow is «split» by the potential barrier petal (see Fi g. 1 1 , bottom right) into two flows. One flow moves diagonally to the right, whilst the other moves vertically upwards towards the cut in the region (see Fig. 8 , top right). The flow entering the upper part of the region fr om the split is attracted towards the centre by the potential well, an d then encounters the potential barrier, which directs it diagonally towards the exit of the re gion. As in the previous case, the principal density 2,2 f is concentrated on the perimeter of the region (see Fig. 8 , bottom right, red). To conclude this section, let u s consider a special case of th e simplest factorised solution for 0 λ = and R const = . Theorem 7. Let 2 >  , 0 λ = and R const = then the factorised solution () 0 u λ θ = =Θ of equation (1.12) corresponds to the solution () , ,, n rt φ ψ  of the Schrödinger equation () () ( ) () 2 ,, , , ,, ,, ,, , 2 tn rn n n ir t r t U r r t m φφ φ ∂= − Δ+ ψψ ψ      (3.18 ) () 2 , 2 2 , 11 E ,, e x p , 22 n T rr r n n n n rt N n i n it rr σσ ρ σ φφ       ++  =− + −   ψ       (3.19) 2 2 , 1 12 , 3 , 4, ... 2 r n n N nn n πσ − +−    =Γ =          (3.20) where the potential () , n Ur  and the quantum potential () , Q n r  take the form () () ( ) () 2 2 2 2 , 22 21 1 Q, 8 nn rr n nn n r m rrr σσ  ++ + =− − +         (3.21) () () ( ) () 2 2 2 22 2 , 22 21 1 E, 8 nn rr nT r nn n Ur mr r r σσ ρσ  ++ +  =− − + − +        (3.22) and satisfy the Hamilton-Jacobi equation (i.10) with rv e v r φ σσ =   . The constan t 1 c i n t h e definition (1.20) of the function () 0 θ Θ is taken to be equal to 1 T r c ρ σ =− . The proof of Theorem 7 is given in Appendix B . Remark 7. It follows from a comparison of expressions (3.17) and (3.21) t hat E0 = . Despite the singularity at the point 0 r = , the limit 0 lim 0 r → = ψ since the exponential function tends to zero faster than the power function. Thus, expression (3.19) for the wave function can be defined at 28 the singular point as () 0, , 0 t φ = ψ . Note that the asymptotic behaviour of the probability density , ~ n f r −   when r →∞ , so t he condition 2 >  is necessary for the convergence of the normalisation integral , 0 2 n f rdr π ∞   . Theorem 7 gives an explicit expression , n N  only when 3 , 4, 5, ... =  although the normalisation integr al exists for any finite 2 >  . Consider t he form of the potential (3.22) in which there is a t e r m w i t h t h e c o e f f i c i e n t 22 T r ρ σ . A similar coefficient 2 T rv r m σσ ρ σ =  is present in the probability flux v  . Depending on the values of the coefficient 1 T r c ρ σ = , three variants of the potential energy are possible () , n Ur  () () () () 22 2 2 22 2 2 , 2 22 2 0, 2 , 1 0, 2 , ~ 1 , , 1 0, 2 , T rr p n Tn rr p T rr p r Ur r f o r r r ρσ σ σ ρσ σ σ ρσ σ σ +  −< <  +    −=  =  −→ + ∞     − −> >           ( 3 . 2 3 ) where def pv m σσ = is the characteristic length in t erms of momentum . The variant s of (3.23) differ in the behaviour of the potential (3.22) over large dist ances r as well as in the presence of different numbers of zeros () , :0 , 1 , 2 jn j rU r j ==  () () () () , 22 2 , 22 2 , 22 2 1 ,2 , , 2 1, 2 , 2 0 , ,2 . rp n n n n j nT rr p r T r n n rp T r n nD rD n n Dn σσ σσ σ ρ σ ρσ σσ ρσ   <  +   −− ±  =+ = = + + >  −    −− >  −                ( 3 . 2 4 ) F i g . 1 2 shows the radial distributions of the potentials , n U  on the left, and the corresponding (by colour) probability dens ity distributions 2 ,, nn f = ψ  in the centre, and the vector field of the probability flux v  on the right. All graphs in Fig. 1 2 are plotted for the values 4 n = and 6 =  . The values of the characteristic coordinate dim ensions () , 1...3 k r k σ = are taken based on the three conditions (3.23)-(3.24). For all valu es of () , 1...3 k r k σ = t h e p o t e n t i a l 4,6 U has a pole at the origin (infinite potential barrier) so all p robability densities 4,6 f h a v e maxima at certain radii outside the central region. The m axima of the densities 4,6 f are located in finite potential wells. For () 1 r σ ( i n F i g . 1 2 , l e f t a n d c e n t r e ) , t h e p o t e n t i a l 4,6 U h a s t w o z e r o s 1 r and 2 r (3.24). The maximum probability (the area under the function 4,6 f ) is localised within 12 rr r << (red curve in Fig. 1 2 , centre). The potentials for the values () 2 r σ (blue graph) and () 3 r σ 29 (green graph) each have a single zero 1 r (3.24), but different orders of asymptotics at r →∞ (3.23) – 6 ~1 r − and 2 ~1 r − respectively. The vector field of the probability f lux (Fig. 1 2 , right) has a vortex structure rv e v r φ σσ =   . The white arrow on the right in Fig. 1 2 indicates the characteristic direction of the flux, which is anticlockwise. The velocity v of the probability flow is maximum in the central region (red in Fig. 1 2 , right) and minimum (blue in Fig. 1 2 , right) at a distance from it. A comparison of the graphs 4,6 f and v in Fig. 12 shows that in the vicinity of the pole ( 0 r = ) the probability density 4,6 f is negligibly small. For this reason, the circular region has been cropped out of the right-hand side of Fig. 1 2 of radius 1 r (the first zero (3.24) of the potential 4,6 U at () 1 r σ ). Note that the field v  according to (1.3) is potential, however curl r v θ ≠   . The problem is that the potential Φ (the phase of the wave function 2 ϕ =Φ ) is not a smooth function () 1 , xy с φ Φ= − , i.e. there is a discontinuity at 0 φ = and 2 φπ = . This peculiarity of the wave function’s phase leads to the presence of a vortex in Fig. 12 (right). A consequence of this vortex is the natural fulfilmen t of the Bohr-Sommerfeld quantis ation rule for momentum pm v =  0 1 ,2 , 2 h pd l c h γ π ==      ( 3.25) where the integration is perfo rm ed over the closed contour 0 γ , containing the orig in (the velocity pole 0 r = ). Since 1 0 c > it can act as a quantum number s , for example, 1 2 cs = . In this case, expression (3.25) coincides exac tly with the Bohr-Sommerfeld qu antisation rule. Remark 8. In the general case, i nstead of the potential expansion (1.3) f or representing the field v  , one can use the full Helmholtz decomposition (i.7), containin g the vortex component A γ  . In this case, instead of the Schrödinger equation (3.18), the electromagnetic equation (i.9) applies. From a physical point of view, there is only one quant um system with a vortex field v  . Describing it using the Schrödinger equation (3.18) or (i.9) is equivalent, and is related to the Fig. 12 Potential energy, probability density and probability flux dens ity for the quantum system (3.18)-(3.22) 30 gauge invariance of the Schrödinger equation itself. This issue is discussed in detail in [ 64 ], which also shows that for t he field v  there exists positive calibration-invariant t he Weyl- Stratonovich function of the el ectromagnetic quantum system. No te that the solution (3.19) is a special case of the solution for the so-called Ψ -model of micro- and m acro-systems [ 61 ]. Note that the value 1 c is an arbitrary constant when constructing the solution for th e angular part (1.20). In the formulation of Theorem 7, the value 1 c is formally redefined in terms of another constant r σ , that is 1 T r c ρ σ =− . Looking at the inequalities on the right-hand sides of systems (3.23)-(3.24), one is reminded of a well-known inequ ality in quantum mechanics – the Heisenberg uncertainty principle 2 rp σσ ≥  . A natural question arises regarding the relationship between these inequalities. Recall that in §1, the quantity v σ was i ntroduced as a characteristic length in mom entum space (1.11) for the size of the ellipticity region ( T ρρ < ) of equation (1.8) for the phase of the wave function. The quantity r σ is a certain characteristic length in coordinate space, app earing in t he expression for the potential , n U  and the velocity v  . Nevertheless, there is a dire ct link between the quantities r σ a n d r σ . Indeed, given the probability density function , n f  , one can calculate r σ (see Appendix B ) 2 22 , r rr σ =− 1 2 12 ,5 , 6 , . . 4 . 3 2 1 n r r nn n n n σσ −       =Γ Γ − Γ =       −        Γ   +−−       ( 3 . 2 6 ) A similar procedure can be c arried out for the quantity v σ by moving from the coordinate wave function (3.19) to the momentum function, and c alculating p σ . The quantity v σ associated with the ellipticity region (1.11) is a parameter i n the initial distribution , n F  (i.18) and allows conditions to be imposed on it. Conclusion Within the Wigner-Vlasov formalism, the exact solutions for co ordinate space obtained in this work can be extended to phase space. Indeed, knowledge o f t h e w a v e f u n c t i o n () ( ) () ,e x p , rt f r i rt ϕ =   ψ   allows one to find the Wigner function () () () () () 2 *2 1 2 ,, , e x p Wr p rt r t p s d s π −+ =− ⋅  ψψ        , where 2 rr s ± =±   . The function W satisfies t he Moyal equation or the second Vlasov equation (i.2 ) with the Vlasov-Moyal approximation (i.15). Note that when deriving the chain of Vlas ov equations, no condition was imposed regarding the non-positivity of the functions n f . Therefore, the presence of negative values i n the quasi-proba bility density function W does not contradict the chain of Vlasov equations. 31 Appendix A Proof of Theorem 1 It follows from the theory of differential equations that the c haracteristic function is a solution to the following equation () () 12 , 1, 2 , 11 ,, n n a a dd d d λρ θ ρ ρ ρ ρ θ θ ±Δ ==  ±= Δ   (A.1) where, for equation (1.12), 11 12 1, 0 aa == and (1.13) is taken into account. For the hyperbolic equation T ρρ > an d () , 0 n ρ Δ>  . Therefore, the integration of e xpression (A.1) takes the form () , 2 2 1 11 1 2 1 1 n n n T n dd d t d t const nt ρρ ρ θρ ρ ρρ ρ ρ ρ ± = Δ= + + += − = + −= +       () ,, 22 arctg a , 11 1 rct 1 g nn tt nn  = ΔΔ ++ − ++ −=       ( A . 2 ) where 1 T ρρ ρ => , 1 n t ρ =− . F r o m e x p r e s s i o n ( A . 2 ) i t f o l l o w s t h a t ( 1 . 1 4 ) h o l d s i n t h e hyperbolic region T ρρ > of e quation (1.12). In t he elliptic region T ρρ < , the determinant is () , 0 n ρ Δ<  and equation (A.1) has an i maginary right-hand side ,, nn i ±Δ = ± − Δ  . The integration of such an equation is analogous to the procedure i n (A.2) () , 2 2 11 2 11 1 1 n n n n T dd d t d t const nt ρρ ρ ρρ ρρ ρ θρ ρ ± = −Δ = + − + + += = − = − −       () ,, 11 1 22 arcth arct h , 1 nn tt nn  =  −Δ −Δ ++ −  + −=    +    ( A . 3 ) where 1 n t ρ =− . Expression (A.3) proves the validity of (1.15). When T ρρ = (the parabolic region of the equation), both characteristics (A .2) and (A.3) c oincide. Let us reduce equation (1.12) to canonical form. Using the representatio n (A.2) for th e hyperbolic equation, we obtain () () () , 1 ,, ,, 1 2 , 1 11 n nn n n TT n nn n n T Tn n h n n ρρ ρ ρρ ρ ρ ρρ ρ χ ρρ ± −  ∂  =∂ =  ++  − −Δ = −  ΔΔ        () () () , , , , , n h nn ρ ρ ρ χ ρ ± ∂ Δ == Δ    ( A . 4 ) () () () ( ) 2 , , 2 , , 2 ,, . 12 21 22 n h n n n T n nn n n n ρ ρ ρ ρ χ ρ ± ∂ + −Δ −Δ + + = ΔΔ =       ( A . 5 ) The partial derivatives of equation (1.12) take the form () () () () , u ρρ ρ χχ χχ +− +− + =Ω Ω 32 () () () () () () () () () () () () () () 22 2, u χ ρ ρ ρ ρ ρ ρρ ρρ χχ χχ χ χ ρ χ χχ χ χ χ χ ++ +− −− + − ++ − − + −   ++ Ω + Ω+  Ω  ΩΩ  = ( A . 6 ) () () () () () () . 2 u χ θθ χχ χχ χ ++ +− −− =Ω + Ω− Ω Substituting expressions (A.4)-(A.6) into equatio n (1.12), we obtain () () () () () ( ) () () () ( ) () () () () () () () () 2 , ,, , 2 2 , , 2 2 12 0 2 n n n n n n nn χχ χχ χχ χ χ χχ χ χ χ χχ χ ρ ρ ρρ ρ ++ +− −− + − +− + + + −− − Δ+ + −  =Ω Ω Ω Ω Ω −   Δ = Δ  ++ + + Δ    −Ω Ω −Ω − Ω +Ω      ΔΔ +           () () () () () ( ) () () () ( ) () () 2 , , , 22 , ,, 2 , , , 2 12 2 12 2 , 2 1 4 4 n n n n nn n n n n n n n χχ χ χ χχ χ χ ρρ ρ ρ +− + − +− +−   Δ+ + −  ΩΩ Ω − Δ =     Δ  Δ =+ + Δ +    Δ+ + − Δ  =Ω Ω Ω   Δ Δ   − +            () () () ( ) () ( ) 2 , 32 , , 12 2 0. 8 n n n n n χχ χ χ +− +− ++ − Δ  ΩΩ Ω =   Δ− ++ Δ     ( A . 7 ) Equation (A.7) coincides with the second equation in (1.16). T aking (A.3) into account, we perform sim ilar transform ations for the elliptic equation () , , , , n e n ρ χ ρ ± −Δ ∂=   () () ( ) , , 2 2 , , 21 2 , n n n e nn ρ χ ρ ± − ∂ Δ+ + =− −Δ     ( A . 8 ) Substituting (A.8) and ( A.6) into equation (1.12 ), we obtain () () () () () () () () () ( ) () () () () () () () () 2 , , 2 , , 2 ,, 21 0 2 2 2 n n n n n n n n χχ χ χ χχ χ χ χ χ χχ χχ χχ ρ ρ ρρ ρ ++ + − −− + − + − ++ +− −− Δ− + +  =− Ω Ω Ω Ω Ω −   − = Δ Ω  ++ − + Δ −Δ    Δ   −Ω −Ω − Ω Ω      Δ ++        () () () () () ( ) () () 2 2 , , 2 , , 22 1 2 , 2 n n nn n n χχ χχ χ χ ρ ρ ++ −− +− Δ− + ΔΔ  +− +   = ++   −Ω Ω Ω Ω     −  Δ      () () () () () ( ) () () () ,, 32 2 , 21 2 0. 4 nn n n n χχ χχ χ χ ++ −− +− − Ω ΔΔ +− + Ω Δ ++ +  ΩΩ =   −    ( A . 9 ) Equation (A.9) coincides with t he first equation in (1.16). Th eorem 1 is proved. Proof of Theorem 2 Let the variable ρ be expressed in terms of the variable ς , then equation (1.19) for the function R takes the form ,, RR R R R ρς ρ ρ ρς ρ ρ ς ς ρ ρ ςς ς ς ′′ ′ ′ ′ ′ ′ ′ ′ ′ ′ == + 33 2 ,, 2 0. nn gg RR R ς ς ρρ ρ ρ ρ ς λ ςς ς ς ρρ  ′′ ′ ′ ′ ′ ′ ++ − =    ( A . 1 0 ) We require that the condition () () () , , 0l n 1 1 1 l n , n n n n g dd g const n ρρ ρ ρρ ρ ς ς ς ρρ ρρ ρ  ′′ ′ +=  =− = + − = + − +       () 1 1 00 1 1 exp , n n n T n T e cc n ρ ρρ ςρ ρρ ρ + + +   + ==         ( A . 1 1 ) where the substitution ρ ςς ′ = a n d T ρρ ρ = has been made. Using expression ( A.11), we find the variable ς () 1 ,0 1 . n n nT d ce ρ ρ ς ρρ ρ + + =     ( A . 1 2 ) When condition (A.12) is satisfie d, equation (A.10) takes the f orm () () 2 2 1 2 , 2 22 2 2 0 1 01 , n n n n T g R RR e R c ρ ςς ςς λ λ ρρ ρς ρ + − + ′′ ′′ =− =+ −     ( A . 1 3 ) which proves the validity of e xpressions (1.21)-(1.22). Let us evaluate the integra l (A.12) for two cases: kn =  and kn ≠  , 0, 1 , 2. . . k = Without loss of generality, let us begin with t he case kn =  . For convenience of transformati on, let us intr oduce the notat ion () () () , def ,, 0 , 1 . n nk x nk n T nk kn e Jx d x c J x β ς ρρ ρ + =  =   ( A . 1 4 ) Let us compute the derivative of the integral exponential func tion (1.25) and take into account expression (A.14) () 1 11 0 11 1 Ei 1 , !! ! kk k x kk k x xx e x x kx k x k x − +∞ +∞ +∞ == =  ′ =+ = + = =     () () () , ,, 0 , 0 1 Ei Ei . n nk x nn nk n n ne x Jx x x n β ββ ′ =  = ( A . 1 5 ) Note that the follo wing relation holds () () , , ,, , , , 11 11 1 . n n nn n nk nk nk nk nk x n xx x x nk nk nk nk nk nk kn ex n n k ee e e xx x k n k x x β ββ β β β β ++ −+ ′′   − =  =−       ( A . 1 6 ) Let us integrate expressi on (A.16) and take into account the n otation in (A.14) 34 () () ,, , , 11 1 . nn nk nk x x nk nk kn kn ed x e Jx kk n x x ββ β −+ =−  ( A . 1 7 ) We transform the integral in expression (A.17) i nto the form ,1 nk J − by performing a substitution ,1 , n nk nk xx ββ − = () () () ,, 1 11 ,, ,1 , ,1 11 11 ,1 ,1 . nn nk nk kk xx nk nk n nk nk nk kn kn nk nk ed x e d x Jx xx ββ ββ ββ ββ − −− −− −+ −+ −−   ==        ( A . 1 8 ) Substituting (A.18) into expression (A.17), we obtain () () , , ,, 1 , , 1 1 ,1 1 . n nk k x nk n nk nk nk nk kk n nk e Jx J x kk n x β β ββ β −− − − =− ( A . 1 9 ) Expressions (A.15), (A.19) and (A.14) prove the validity of ex pressions (1.23) and (1.24) for kn =  . Let us evaluate the integral (A.12) for kn ≠  . In this case, the direct expansion of the exponential into a series and its integration term by term yiel ds the expression () 1 1 1 00 11 1 , !! n kk nk nk n kk d ed c o n s t nk n n k k ρ ρρ ρρ ρ + − +∞ +∞ −− + == ++   == +   −           ( A . 2 0 ) which corresponds to expression (1.23) for kn ≠  . Theorem 2 is proved. Proof of Theorem 3 Let us make a change of variable s in the hyperbolic region of t he equation () () () () () ( ) ( ) ,, , , ,, , , , , , 1 ,, , , 2 hh h h nn n n n n n μχ χ χ χ μ μ ±+ − + − + −      =± Ω = Ω             ( A . 2 1 ) then equation (1.16) takes the form () () () ( ) () () ,, ,, , ,, , 11 11 ,, 22 22 hh nn n nn n χμ μ χμ μ ++ − −+ − Ω= Ω + Ω Ω= Ω − Ω       () () () () () () () () () () () () () () ,, , , , , , , ,, ,, ,, ,, 1111 1 , 4444 4 hh n n nn nn nn nn nn nn χ χ μμ μμ μμ μμ μμ μμ + − ++ +− −+ −− ++ −−   Ω = Ω− Ω+ Ω− Ω = Ω− Ω                () () () () () () ,, ,, , , 0. 4 nn nn n h n μμ μμ μ κ ++ −− + Ω− Ω Ω = +       ( A . 2 2 ) We shall seek the solution to equation (A.22) in the form () , n μ +  Ω=Ω    , then the e quation for the function Ω takes the form () () def ,, . 44 00 , hh nn κκ ′′ ′ ′ ′ Ω+ Ω =  ΛΛ = Λ = Ω +  ( A . 2 3 ) We shall find the solution to equation (A.23), taking into acc ount the form of the coefficient () , h n κ  (1.17) 35 ( ) () ( ) () ,, ,1 , exp 4 , 4 h n h n nn d d с d κμ κμ ++ Λ   =−  Λ= −   Λ     ( A . 2 4 ) where 1 с is a constant. Let us evaluate the integral (A.24) using expre ssions (A.21) and ( 1.14). Let us introduce the substitu tion () () , , 1 2 arctg , 1 1 11 n n nn n n n nn dd n d n ρ ε ε μ εε ε ε + Δ + == = + + =−  − +     ( A . 2 5 ) hence () () () () ( ) () ( ) () () () () ( ) () () 3 2 , , 32 , , 2 2 2 32 2 , 81 81 22 l n 2 l n 1 1 , 1 81 12 2 1 12 1 2 8 1 1 1 22 n n n nn n h n n n n n n n n nn n n nn n n n n d n d n n d n nn d nn n n εε ε ε κ ε ρμ ε ε ε ε ε εε ε ε ε ε ε ε + ++ − Δ + Δ ++ − + + + + Δ− == + − == + −− − + + + == + +− +                   () ( ) () () 4 1 , 8 1 ln 1 . 4 n h nn n n n d n ε κε ε μ + + =− + +     ( A . 2 6 ) Substituting the integral (A.26) into equation (A.24), we find the function Λ a n d f r o m this, according to ( A.23), the function Ω () () () 2 1 1 1 ln exp , 1 n n nn n d с d n ε εε μ + + −+   Ω =Λ =       ( A . 2 7 ) hence () () 1 11 1 1 1 1 2 , 1 1 11 n n n n n nn n n n n n n n dd nn ee сс e ε ε ε εε εε εε + + + − + ++ Ω + == +        ( A . 2 8 ) where (A.25) has been taken into account and the substitution 1 nn εε =+ h a s b e e n m a d e . T h e integral in expression (A.28) is similar to the integral (A.12) . Let us consider the case where , 0,1 , 2... kn k ==  . By analogy with expressions ( A.15)–(A.17), we obtain () , def , 1 , nk n nk n n k n e I d βε εε ε + =  () () () ,0 ,0 ,0 ,0 Ei Ei , nn nn n n nn nn de I d βε β εε β ε εε =  = ( A . 2 9 ) ,, , , , ,, 11 1 , nk n nk n nk n nk n nk n nn k n k kk k k k nn n n n nn k de e e de e dk k d βε βε βε βε βε εβ β εε ε ε ε εε ++   − =  =−     () , , , , 1 . nk n nk n nk n nk n kk nn ed e I kk βε βε β ε ε εε =−  ( A . 3 0 ) We transform the integral in exp ression (A.30) 36 ,, 1 11 ,, , , 1 ,1 11 ,1 ,1 ,1 , ,. nk n nk n kk nk nk nk nk nn nk n n n kk n nk n nk nk nk ed e d I βε β ε ββ β β εε εε ε εβ ε β β β − −− − − −+ −− −     == =                  ( A . 3 1 ) Substituting (A.31) into expression (A.30), we obtain the re cu rrence relation () , ,, ,, 1 1 ,1 ,1 1 . nk n k nk nk nk n nk n kk nk nk n e II kk βε ββ εε ββ ε − − −−  =−    ( A . 3 2 ) Expressions (A.29), (A.32) and (A.28) prove the validity of ex pressions (2.1)-(2.3) for kn =  . Let us evaluate the integral (A.28) for kn ≠  . In this case, the direct expansion of the exponential into a series and its integration term by term yiel ds the expression () 1 1 2 1 00 11 1 , !! n kk kn kn n n n nn n kk n d ed с nk n kn k ε εε εε ε + − +∞ +∞ −− + == ++   == +   −           ( A . 3 3 ) which corresponds to expression (2.1) when kn ≠  . Theorem 3 is proved. Appendix B Proof of Theorem 4 We shall seek a solution to equation (2.5) in accordance with expressions (1.26) and (1.28) in the form () () ,, s RT υ ρρ ττ δ ρ == () ( ) ( ) d , ef , 1, 1 ns n n g γτ τ ρ δ  =+ −    =           (B.1) where ,, s δυ are certain numbers to be determined. Substituting expression (B.1) into equation (2.5) 11 , s R Ts T υυ ρ τ υρ δ ρ −+ − ′′ =+ () () 22 2 2 2 2 12 1, ss R Ts s T s T υυ υ ρρ ττ τ υυ ρ δ ρ υ δ ρ −+ − + − ′′ ′ ′′ =− + + −+ (B.2) () () () 2 2 , 22 , 22 2 1 21 0 , ss n n sT s s T T ττ τ υλγ υ υ ρρ δδ υ γ ρρ ρ −+ − ′′ ′ ++ − + + =   () () () 2 , , 2 1 1 21 0 . n n Ts T T ss ττ τ υλ γ υ υ τυ γ τ −+ − ′′ ′ ++ − + + =   ( B . 3 ) We impose a condition on the coefficient of the function T in equation (B.3) () () ( ) () () () () 22 2 2 2 , 11 1 1 , ns n ττ υλ γ υ υ υ υλ υλ υλ δδ  −+ − = + − + − − + = − − +       () 22 ,1 0 , sn υυ λ =+ − + =  ( B . 4 ) where the roots of equation (B.4) are the numbers () , λ υ ±  (2.7). Satisfying the conditions in (B.4) reduces equation (B.3) to the form 37 () () () () 2 2 , , 1 2 11 11 0 , TT T nn n n τ λ λ ττ υλ υ τ τ δδ ± ± −+  +  ′′ ′ ++ − + − − =          () () () () , 2 2 , 1 2 1 0. n TT T nn n λ ττ λ τ υλ υ ττ δδ ± ± −+  ++ + ′′ ′ +− − =         ( B . 5 ) Let us determine the last parameter () 1 n δ =+  . As a result, equation (B.5) will become (2.6). The resulting equation (2.6) is the Kummer equation and a solution (2.8) is known for it. Theorem 4 is proved. Proof of Lemma 1 The direct solution of the equation () , ak λ + =−  leads to the expression () 2 22 2 2, 44 kn λλ λ += + +−   ( B . 6 ) () 2 2 2 , kn kn λλ −− =  ( B . 7 ) where 2 2 kn λ ≤+  is taken into account w hen squaring equation (B.6). Substituti ng the value  from equation (B.7) into the inequa lity yields the condition () 22 2 2 1 kn λλ ≤− . ( B . 8 ) Note that for the pe riodic boundary conditions (1.20), we have 1 λ ≥ . Thus, expression (2.12) holds. We shall prove that 0 α ≠ by contradiction. Suppose 0 α = , then () () () () , 2 ,, , 2 2 10 2 1 0 , 4 n b n λ λλ υ αυ λ + ++ + =− + + ==  += =      (B.9) It follows from expressions (B.9) and (B.6) that () 2 20 . kn λ +− =  ( B . 1 0 ) Consider the special cas e of the equality (B.8) 42 2 2 2 2 2 02 , 11 4 nn kk λλ λ =+ + −− =  ( B . 1 1 ) where 1 λ ≥ by the condition of the lemma. Substituting (B.11) into expres s i o n ( B . 1 0 ) y i e l d s a violation of the lemma’s condition 1 >−  , that is ( ) 22 22 . 21 1 4 1 1 4 2 kn kn kn kn =− + + >−  +< − +  ( B . 1 2 ) Inequality (B.12) is false. The factor 0 c follows directly from the connection rela tion 38 () () ( ) () () () () () () () () () 1 11 1 ,, , 1 , . 1 b ak ab k M ab z L z M k z L z ba k α α α α − − Γ− Γ Γ+ Γ + =  −+ = Γ− Γ + + ( B . 1 3 ) Lemma 1 is proved. Proof of Theorem 5 Let us write the Legendre transf ormation (1.7) in polar coordi nates and use the factorised representation of the solution () ( ) ~ uR ρ θ Θ sin sin cos c os cos sin , RR xu u R R R ρθ θθ θθ ρ θθ ρρ ρ ′′ ΘΘ  ′′ ′ ′ =− = Θ − Θ = −  Θ  ( B . 1 4 ) cos cos sin sin sin cos RR yu u R R R ρθ θθ θθ ρ θθ ρρ ρ ′′ ΘΘ  ′′ ′ ′ =+ = Θ + Θ = +  Θ  , (B.15) () ( ) ,, 1 . R xy u u R R ρ ρρ θ ρ ′  ′ Φ= − = Θ−   ( B . 1 6 ) Let us compute the deriv ative R ′ for the solution rest ricted to the origin () ( ) () () ( ) () () 11 1 ,, ,, 1 ,, ,, 1 , ,, nn Ma b R M ab M ab M ab Ma b τ υυ υ ρτ τ υρ τ ρ τ ρ τ υ ρ τ −+ − −   ′ ′′ =+ + = + +      () () 1 ,, ,, T RM a b n a b υ ρ ρ τυ τ τ ρ − ′ =+     () ,, . R na b R ρ ρ υτ τ ′ =+  ( B . 1 7 ) Substituting (B.17) into (B.14)-(B.16) yields the validity of expressions (3.1)-(3.3). We find the expression for the Jacobian of the Legendre transforma tion using formula (1.7). Let us calculate the partial derivatives for th e factorised solution () 22 2 2 2 2 2 sin cos sin 2 sin cos sin 2 , RR gg R ξξ ω ρ θθθ λ θ λ θθ ρ ′ Θ  =− − ϒ − + + ϒ   2 2 11 cos 2 sin 2 sin 2 cos 2 , 22 RR g g R ξη ω ρ θθ λ θ θ ρ ′ Θ  ++   =ϒ − + − ϒ     ( B . 1 8 ) () 22 2 2 2 2 2 cos sin sin 2 cos sin sin 2 , RR gg R ηη ω ρ θθ θ λ θ λ θ θ ρ ′ Θ  =− + ϒ − + − ϒ   where it is noted that the functions R a n d Θ satisfy the equations (1.19), i.e. () 22 RR g R R ρλ ρ ′′ ′ =− a nd 2 λ ′′ ΘΘ = − . Substituting the expressions (B.18) i nto (1.7) and performing intermediate calculations, we obtain 2 4 1 2 , RR JA B C uR R ρ ρρ − ′′  =+ +   ( B . 1 9 ) where () ( ) () () 22 2 2 2 cos sin 2 sin sin cos sin 2 1 2c o s 2 1 s i n 2 , 4 Ag g g θθ θ θθθ θθ =+ ϒ − − − ϒ − −ϒ − + ( B . 2 0 ) 39 () ( ) () ( ) () () 22 2 2 2 22 2 2 2 2 sin cos sin 2 cos sin sin 2 cos sin 2 sin cos sin sin 2 1 1s i n 2 2 c o s 2 2 c o s 2 1s i n 2 , 2 Bg g gg gg λθ θ θ θ θ θ θθ θ λθ θ θ λθ θ θ θ  = − −ϒ − + −ϒ +    ++ ϒ − − + ϒ−    −+ − ϒ ϒ − +    ( B . 2 1 ) () () () 22 2 2 2 2 2 2 sin cos si n 2 cos sin sin 2 1 1s i n 2 2 c o s 2 . 4 Cg g g λθ θ θ λ θ θ θ λθ θ   =− − ϒ − + ϒ −    −+ − ϒ  ( B . 2 2 ) Let us simplify the expressions for th e coefficients (B.20)-(B .22). () ( ) ( ) () ( ) 2 22 2 2 2 2 2 2 2 22 2 2 2 2 2 2 2 sin cos cos 2 si n 2 cos sin sin c os sin cos cos sin sin c os sin 2 sin cos cos 2 sin 2 cos 2 , Ag g g θ θ θ θ θ θ θθ θθ θθ θ θ θ θ θ θ θ θ =ϒ − + − + + − + +−+ ϒ − + − ϒ + () 2 . Ag =− + ϒ ( B . 2 3 ) 22 2 2 2 4 2 4 2 2 2 2 2 22 2 2 2 2 4 22 2 2 2 2 2 2 2 2 2 2 4 sin cos cos sin sin cos sin sin 2 sin 2 cos sin 2 cos sin 2 sin sin 2 sin 2 cos cos sin sin 2 cos sin 2 sin sin cos sin sin 2 Bg g g g gg gg g λθ θ λ θ λ θ λ θ θ λ θ θ λθ θ θ θ θ θ θ θ λθ λ θ θ λ θθ λ θ θ λ θθ λ θ θ =− + + − − ϒ + +ϒ + ϒ − ϒ + ϒ + − −+ ϒ − ϒ − + + +ϒ () () 22 2 2 2 2 2 2 2 22 2 22 2 22 cos sin 2 sin 2 sin 1 sin 2 cos 2 2 cos 2 2 sin cos 4 sin c os 2 sin cos 1 cos 2 sin 2 gg gg g θθ θ θ λ θ θ θ λθ θ λ θ θ λ θ θ θ θ +ϒ −ϒ − ϒ + + ϒ + ++ + − ϒ + = () ( ) () ( ) () () () 22 2 2 2 2 2 4 4 2 2 22 2 2 2 2 2 2 22 2 2 2 22 2 2 2 sin cos 2 sin cos 2 cos si n 2 sin cos 2 sin cos 2 sin cos cos sin cos 2 sin 2 sin 2 cos sin cos 2 si n 2 cos s in cos 2 si n 2 cos sin cos 2 2 sin 2 gg g g λθ θ θ θ λ θ θ θ θ λθ θ θ θ λ θ θ θ θ λθ θ θ θ θ θ θ θ θθ θ θ θ =− + + + + + +− + + ϒ + − − + +ϒ − − + ϒ − − + +ϒ − − + ϒ () 2 cos 2 , θ + () 22 2. Bg λ =+ ϒ ( B . 2 4 ) 42 2 2 42 2 2 4 2 2 4 2 2 42 2 4 2 2 4 2 2 2 2 22 2 2 2 2 2 22 2 cos sin s in co s cos cos s in sin 2 sin cos sin cos sin c os sin 2 sin sin 2 cos c os 2 sin 2 sin 2 cos si n 2 sin cos 2 sin 2 sin 2 С ggg g gg gg λ θ θλ θ θλ θ θλ θ θ λθ θ λ θ θ λ θ θ λ θ θ λ θ θλ θ θ λ θ θλ θ θ λθ θ θ =− − − − −+ − + ϒ − −ϒ +ϒ −ϒ +ϒ + +ϒ − ϒ − 22 cos 2 θ ϒ= () ( ) () ( ) 44 4 2 2 2 2 2 22 2 2 2 2 cos sin 2 sin cos sin 2 cos sin cos 2 sin 2 cos 2 cos sin sin 2 cos 2 , gg λθ θ θ θ λ θ θ θ θ λθ θ θ θ θ θ =− + + − ϒ − − − −ϒ − + − − ϒ + () 42 . С g λ =− + ϒ ( B . 2 5 ) Substituting expressions (B.23)-(B.25) into (B .19) gives () () () 2 4 14 2 2 2 2 2 2, RR Jg g g uR R ρ λρ λρ − ′′  −= + ϒ − + ϒ + + ϒ   ( B . 2 6 ) 40 which, taking (B.17) into account, reduces t o (3.4). Let us con sider the special case of ( B.26) for T ρρ = and e θθ = () () ( ) () () () () ( ) () () ( ) 2 12 2 4 22 22 22 0 44 , ,2 1 , 11 0 , Te Te e T T T Te T Te T TT u J uR ρθ ρθ θ ρ ρ ρ ρθ ρ θρ θ ρ ρρ −  =− ϒ − + =  ′ =− ϒ − =− Θ − =       (B.27) where it is assumed that () 0 e θ ′ Θ= . It follows from (B.27) that (3.5) holds. In the case where 1 λ = , we have () ,1 1 υ + =  and () ,, 1 0 n a + =  . Consequently, () 0, , 1 Mb τ = and () 0, , 0 Mb τ τ ′ = , that is () () ,, 1 , 1 1 n ρυ + ==   . ( B . 2 8 ) Substituting (B.28) i nto (B.16) and then into (B.17), we obtai n that 0 Φ≡ , whilst expression (B.26) takes the form () 4 12 2 2 2 20 , Jg g g u ρ − − = +ϒ − +ϒ + +ϒ ≡ ( B . 2 9 ) that is the transform is impo ssible. Theorem 5 is proved. Proof of Lemma 2 It follows directly from expressions (B.14)-(B .15) that () () , , cos , si n , , n n n n d d d xy dd d ρρ ρ μ ε θθ ε ρ μ + + Ω ′′ ′ =Ω =Ω Ω =   ( B . 3 0 ) where it is noted that Ω does not depend on the angle θ . Using expressions (A.25), (A.27) and (2.3), we obtain () () 1 1 1 1 1 1 1 1 2 11 , 1 1 n n n n n n n nn n nT T с e ne с e n ερ ρ ε ςρ ερ ρ ρ ρ εε + + + − − + ′ = ++ Ω= = + +       ( B . 3 1 ) where 1 0 1 1 T n c ce ρ + = +   i s a n a r b i t r a r y c o n s t a n t a n d e x p r e s s i o n ( A . 1 1 ) i s t a k e n i n t o a ccount. Substituting (B.31) into (B.30) proves the coordinate tran sform ation (3.10). Note that it also follows from (B.31) and (A.11) that the function Ω is a s o l ut io n to e q u a ti on ( 1 . 1 9) fo r t he r a d i al part () R ρ w h e n 0 λ = , since ρ ςς ′ = and , 0 n g ρρ ρ ςς ρ ′′ ′ +=  . Conse quently, () R ρ Ω= a n d taking into account (B.16) and (B.31), we obtain () () () ,. xy R ρ ρρ ς ρ ρ ′ Φ= Ω − Ω = − ( B . 3 2 ) Equation (B.32) coincides with t he coordinate solution (3.10). Lemma 2 is proved. 41 Proof of Theorem 6 Let us write the expression for the quantum potential (i.10) in terms of the quantity ln Sf = 2 1 Q, 22 rr SS α β  =Δ + ∇   ( B . 3 3 ) () () , 12 ,, , , ,, , ln ln ln ln , 2 , nn nn n nn n n vn Sf F z N c z c z c σ − == = + − =        ( B . 3 4 ) 2 ,, 1 Ql n l n . 22 nn rn r rn r zc z zc z α β  =Δ − Δ + ∇ − ∇     ( B . 3 5 ) Let us transform the terms in expression (B.35) 22 2 2 ,, , 21 22 2 22 2 2 ,, 2 ln ln 2 ln 2, nn n rn r r n r r n r n n rn r n r zc z z c z z c z nz zc n z z c z zz − − ∇− ∇ = ∇ − ∇ ⋅ ∇ + ∇ = =∇ + ∇ − ∇        () 2 2 22 2 2 ,, , 2 ln 2 , r nn n rn rn n z zc z c n z c n z z ∇ ∇− ∇ = − +     ( B . 3 6 ) 2 2 1 ln ln , rr rr rr r zz zz z zz z ∇Δ Δ= ∇ ⋅ ∇= ∇ ⋅ = − ∇ ( B . 3 7 ) 11 1 , nn n n n rr r r r r r r zz n z z n z z n z z −− − Δ= ∇ ⋅ ∇= ∇ ⋅∇ = ∇ ⋅ ∇ + Δ () 2 21 1. nn n rr r zn n z z n z z −− Δ= − ∇ + Δ ( B . 3 8 ) Substitute (B.36)–(B.38) into (B.35) () () () 2 22 2 2 ,, , , 2 21 Q2 1 , 2 r nn n n r nn n n z z cn z cn z c n z cn n z zz β α ∇ Δ  =− + − + − − −         () () ( ) 2 22 2 ,, , 2 21 Q2 1 2 . 2 r nn n r nn n z z cn z cn z cn n z zz β α ∇ Δ  =− + − + − + −       ( B . 3 9 ) Let us calculate the values of r z ∇ and r z Δ , where 22 zv α ρ α ξ η == = + . Using the coordinate transform ation (3.1), we obtain () () cos cos , , sin sin , rr r rr r xX X xr X yY Y yr Y ρθ ρθ φρ θ φρ θ φρ θ φρ θ == + ==      == + ==     ( B . 4 0 ) 22 22 cos , sin . XY XY XY φφ == ++ ( B . 4 1 ) The solution to system (B.40) is cos sin 1 , r YX z YX XY r θθ θρ θ ρ φφ ρ α − ∂ == −∂ sin cos . r XY YX XY ρρ θρ θ ρ φφ θ − = − ( B . 4 2 ) 42 Similarly, for the de rivatives with respect to the a zimuth ang le φ sin , cos , xr X X yr Y Y φρ φ θ φ φρ φ θ φ φρ θ φρ θ =− = +    == +   ( B . 4 3 ) sin cos 1 , YX z r XY X Y θθ φ θρ ρ θ φφ ρ α φ + ∂ == −∂ cos s in . XY r XY X Y ρρ φ ρθ θ ρ φφ θ + = − ( B . 4 4 ) Using equations (B.42), (B.44) and (B.41), we obtain () ( ) () () () 2 22 2 22 22 2 22 2 22 1 , r XY YX YY X X YX zz z rr X Y YX X Y YX X Y θθ θ θ θθ θρ θ ρ θρ θ ρ αα φ −+ + + ∂∂ ∇= + = = ∂∂ +− − () 2 22 2 2 2 . r z XY z XY X Y θθ ρθ θ ρ ρ ∇ + = − ( B . 4 5 ) Thus, for expression (B.39), we need to find r z Δ 22 22 2 11 . r zz z z rr r r φ ∂∂ ∂ Δ= + + ∂∂∂ ( B . 4 6 ) Let us compute the second deri vatives by differentiating syste ms (B.40) and (B.43) 22 22 02 , 02 , rr r r r r r r rr r r r r r r XX X X X YY Y Y Y ρρ ρθ θθ ρ θ ρρ ρθ θθ ρ θ ρρ θθ ρ θ ρρ θ θ ρ θ  =+ + + +   =+ + + +   ( B . 4 7 ) 22 22 2, 2. XX X X X X YY Y Y Y Y ρρ φ ρ θ φφ θθ φρ φ φ θ φφ ρ ρ φ ρ θ φ φθ θ φ ρ φ φθ φ φ ρρ θθ ρ θ ρρ θ θ ρ θ  −= + + + +   −= + + + +   ( B . 4 8 ) Since the values of , rr ρ θ a n d , φφ ρ θ are known from the solutions to (B.42) and (B.44), we shall solve the systems (B.47)-(B.48) for , rr rr ρ θ and , φφ φφ ρ θ . Note t hat to find (B.46), it suffices to know only rr ρ and φφ ρ that is () () () () () () ( ) () () 3 2 22 2 2, rr X Y XY X Y X Y XY X Y XY X YX Y X Y X Y X YX Y X YX Y X Y X Y ρ θ θ ρ ρρ θ θ ρρ θ θ θ θ θθ θ θ ρ ρ ρ θ θθ ρ θ θθ ρ ρ ρ −− + = − − + +− − + − − − ( B . 4 9 ) () () () ( ) () 32 2 X Y XY XY X Y X Y XY X Y XY X X Y Y ρθ θ ρ φ φ θ θ ρθ θ ρ ρ ρθ θ ρ ρ θ θ ρ −= − − − − + + ( B . 5 0 ) () () () () () 2 2. X Y X Y XX YY XX Y Y X Y X Y XX YY ρθ θ θ ρθ θ θ ρ ρ θθ θ θ θθ ρ ρ +− + + − − + Substituting (B.49), (B .50) and (B.42) into (B.46) , we obtain 2 11 1 r rr r z zr r φφ ρρ ρ ρ Δ  =+ + =   43 () () () () () () () () 22 33 22 22 X Y XY X Y XY X Y XY X X Y Y XY X Y X Y XY X Y X Y ρρ θ θ ρρ θ θ ρρ θ θ ρρ θ θ ρθ θ ρ ρθ θ ρ ρρ −− − + =− − − −+ −+ () () () () () () () () 22 33 22 22 XY X Y X Y X Y XY X Y X X Y Y XY X Y X Y XY X Y X Y θθ θ θ θθ ρ ρ θθ θ θ θθ ρ ρ ρθ θ ρ ρθ θ ρ ρρ −− − + −− + −+ −+ () () () () () () () () 22 3 22 22 X Y XY X Y XY XY X Y X Y XY XY X Y X Y XY X Y X Y θ θ ρθ θ ρ θ θ ρθ θ ρ ρθ θ ρ ρθ θ ρ ρ ρ −− − − ++ + −+ −+ () () () () () () () () () () 33 22 22 22 . X YX Y X X Y Y X X Y Y X YX Y X YX Y X Y X Y X Y XY X Y X Y XY X Y ρθ θ θ ρθ θ θ ρ ρ ρθ θ θ ρθ θ θ ρ ρ ρ θ θρ ρ θ θρ ρρ −+ + − − − +− −+ −+ () () () () ( ) () () ( ) () () () () () 3 22 22 22 2 z X Y XY X Y X Y XY X Y XY X X Y Y z XY X Y X Y X Y X X Y Y X Y X Y XX YY XX YY XY X Y X Y XY ρ θ θ ρ ρρ θ θ ρρ θ θ θ θ θθ θ θ θθ ρ ρ ρ ρ ρθ θ θ ρθ θ θ ρ ρ θ θ ρ ρ ρ Δ   −+ = − − − + + −    −− − + + +    +− + + − − − =  () () () () () () () ( ) () 22 22 22 22 22 2 X Y X Y XY XY X Y X Y X Y XY XY X Y X X Y Y X Y ρρ θ θ ρρ θ θ θθ θ θ θθ ρ ρ ρθ θ θ ρθ ρ θ ρ θ = − − + +− − + ++ +− + + () () () () () () ( ) 3 22 22 2. z X Y XY X Y XY X Y X Y XY X Y z XY X Y X X Y Y ρ θ θ ρ ρρ θ θ ρρ θ θ θθ θ θ θθ ρ ρ ρθ θ θ ρθ ρ θ ρ θ ρ Δ −− = − + + − + − −− + (B.51) Before substituting (B.45) and (B.51) into (B.39), it is n eces sary to find the partial derivatives of the functions X and Y . From t he factored solution of the Legendre transformation (3.1), it follows tha t () cos s in , sin cos , , , , RR R XR Y R n a b R θθ θ θ ρ υ τ τ ρρ ′ ′′ ′ ′ =Θ − Θ =Θ + Θ = + =  (B.52) Let us differentiate the expre ssions in (B.52) with respect to the variable ρ () () () () 2 2 2 2 c o ss i n c o ss i n , sin cos cos sin , RR Xg g RR Yg g ρ ρ λθ θ θ θ ρρ λθ θ θ θ ρρ ′ ′′ =Θ + Θ − Θ + Θ ′ ′′ =Θ − Θ + Θ − Θ ( B . 5 3 ) where () 22 Rg R R λ ρρ ′′ ′ =− is taken into account. Redifferentiating (B.54) with respect t o the variables ρ and θ yields the expressions () () 22 2 2 3 22 2 cos sin 2 cos co s 2 sin cos sin 1 cos cos 2 sin , R Xg g g R gg g ρρ λθ λ θ θ λ ρ θ θ ρ θθ λ θ ρ θ θ ρ ′′ ′  =− Θ − Θ +Θ + Θ −Θ +  ′   ′′ ′ +Θ + Θ + + Θ − Θ + Θ   (B.54) 44 () () () () 22 2 2 1 cos 1 sin 1 cos si n , RR Xg g g g ρθ λθ λ θ θ λ θ ρρ ′    ′′ = + Θ− + Θ − + Θ− + Θ    (B.55) () () () () () 2 22 2 22 2 32 3 sin cos cos sin cos sin 2 sin sin cos sin cos sin , RR R Yg g g g RR R R gg g g g ρρ λθ θ θ θ θ θ ρρ ρ θλ θ θ λ θ λ θ θ ρρ ρ ρ ′′ ′ ′′ ′ = Θ − Θ − Θ −Θ − Θ −Θ − ′ ′′ ′ ′ −Θ− Θ− Θ + Θ+ Θ − Θ () () 22 2 22 2 2 3 sin 1 sin cos sin 2 cos sin cos 2 si n sin 2 cos , R Yg g g R gg g ρρ θλ θ θ ρ θ θ ρ λθ λ θ θ λ ρ θ θ ρ ′   ′′ ′ = Θ+ + Θ− Θ − Θ− Θ +   ′′ ′  +− Θ + Θ − Θ + Θ + Θ  (B.56) () () () () 22 2 2 1s i n 1 c o s c o s 1 s i n . RR Yg g g g ρθ λθ λ θ λ θ θ ρρ ′    ′′ =+ Θ + + Θ − + Θ + + Θ    (B.57) Let us differentiate expression (B.52) with respect to the var iable θ 2 cos s in , RR XR R θ θλ θ ρρ    ′′ ′ =− Θ + − Θ       ( B . 5 8 ) 2 sin cos , RR YR R θ θλ θ ρρ    ′′ ′ =− Θ + − Θ       ( B . 5 9 ) () () 22 2 21 c o s 1 2 s i n , RR XR R θθ λλ θ λ θ ρρ    ′′ ′ =− + Θ + + − Θ       (B.60) () () 22 2 21 s i n 2 1 c o s . RR YR R θθ λλ θ λ θ ρρ    ′′ ′ =− + Θ + − + Θ       (B.61) Using (B.53)-(B.61), let us calculate the relations appearing in expressions (B.45) and (B.51). () () 2 2 2 22 2 2 2 4 1, u XY g ρρ λ ρ   += ϒ −+ −      ( B . 6 2 ) () () 2 2 2 22 2 2 2 1, u XY θθ λ ρ   += ϒ − + −      ( B . 6 3 ) () () 2 2 2 22 3 1, u XY X Y g ρθ θ ρ λ ρ   −= − ϒ − + −      ( B . 6 4 ) () () () 2 2 3 11 , u XX Y Y g ρθ ρ θ λ ρ += − − − ϒ  ( B . 6 5 ) () ( ) () () () () 22 2 2 22 2 44 2 22 2 4 21 1, uu XY X Y g g g u g ρρ θ θ ρρ ρλ ρρ λλ ρ ′ −= − + − + −ϒ + +− − + ϒ   (B.66) () () () () () 22 2 2 2 22 2 2 2 22 2 11 , uu u XY X Y θθ θ θ θθ λλ λ ρρ ρ −= − − − − ϒ − + ϒ − −    ( B.67) () () () () () 2 2 2 22 2 3 11 1 , u XY X Y g g ρθ θ θ ρθ λλ λ ρ  −= − − − + − − − ϒ      (B.68) 45 where the representation (B.52) is taken into account  . Using (B.63) and (B.64), we transfor m (B.45) () () () () 2 2 2 22 2 2 22 2 2 22 1 . 1 r z zu g λ ρ λ ϒ− + − ∇ =   ϒ− + −       ( B . 6 9 ) For the convenience of further transformations, let us introdu ce the following notation 2 12 3 4 1, , , , zz z g z λ =− =− = = ϒ  ( B . 7 0 ) and also take into account the exp ression for the derivative g ′ (1.13) () () () 1 11 1 . nn T gn n n g ρ ρρ ρ ρ − ′ =− + =− + = − −   (B.71) Using (B.66), (B.63), (B.68), (B.65), (B.67), (B.62), (B.71) a nd the notation (B.70), we obtain the following relations () () () ( ) () {} () 4 2 2 2 2 22 2 2 22 2 33 2 1 4 1 2 3 4 1 4 2 6 11 2 , u X Y XY X Y z n z n z z z zzz z z z z ρρ θ θ ρρ θ θ λ ρ  −+ = + − + + + + + +   () ( ) () () 4 22 2 2 12 3 4 1 3 2 3 12 6 11 , u X Y X Y X X Y Y z zz z z z z zz z ρθ θ θ ρθ ρ θ ρ θ λ ρ   −+ = − − + −   (B.72) () () () ( ) 4 22 2 2 2 2 2 2 2 2 2 21 4 4 1 2 1 4 3 2 6 . u X YX Y X Y z z z z z z z z z z θθ θ θ θθ ρ ρ λ ρ  −+ = − + + + +  Equations (B.72) appear on the ri ght-hand side of expression ( B.51). The left-hand side of (B.51) is expressed in terms of (B.64). Consequently, (B.51) takes the form () () ( ) () {} () ( ) ( ) () () 2 3 22 2 2 2 22 2 2 22 2 41 32 3 3 2 1 4 1 2 3 4 1 4 2 2 22 2 22 2 22 2 2 2 2 2 2 1 4 4 12 1 4 32 12 3 4 1 3 2 3 12 11 2 21 1 uz z zz z z n z n z z zz z z z z zz z z zz z zz zz z z zz z z z zz z zz λ ρ λλ Δ  += + − + + + + + +     −+ + + + − − − + − =     () ( ) () () 44 3 4 44 3 4 2 222 23 2 14 1 2 3 4 14 1 2 4 3 3 12 4 1 2 3 4 222 3 2 22 2 2222 23 2 322 2 3 42 1 2 4 1 234 1 2 4 1 2 3 4 1 2 4 1 23 4 1 23 2 3 22 2 2 12 3 1 3 2 3 12 4 21 1 2 21 1 z z zzz z z z zzz z n z n z z z zzz z z z z zzz z z z z z z z z zzz zzz z zzz z z zz z z zz z zz z λ λλ λ  =+ − − + + − + + + +  ++ − − − − + − −  −− − + −   () ( ) 42 2 3 2 23 3 1 2 3 11 . zz n z n z z z λ  ++ − + + −   () () () () ( ) () {} () () ( ) {} 2 3 22 2 3 4 4 1 3 2 11 2 3 24 2 22 2 2 2 12 3 3 2 3 1 3 12 4 32 21 3 3 2 3 23 1 3 1 11 1 . uz zz z z z z zz z z z zz z n z n z z z z zzz zz z z z n z n ρ λ λ Δ += + − +  ++ + − + + − −  +− + − + +     (B.73) Expressions (B.73) and (B.69) can be rewritten in a compact fo rm using the notation (3.14) 46 () () () 2 2 2 ,, 1 2 3 4 0 3 2 22 2 14 3 2 ,, , k k n k Az z z z z zu zz z z λ ρ = Δ = +    () 2 22 2 2 14 2 2 22 22 2 14 3 2 . r z zz z zu zz z z ρ ∇ + = +  ( B . 7 4 ) Let us simplify expression (B.34) for the coefficient 1 , n c −  . According to (1.11), we obtain () () 1 ,, 11 1 , 1 n n nn n nT n n T c n cn z g ρ αρ ρ ρ −  == + = + = +    −   +     ,3 11 , n n cn z g z −= − = −   ( B . 7 5 ) () ( ) ( ) ( ) ( ) 2 22 2 ,, 3 3 2 1 2 1 211 2 . nn nn cn z c n n z z z n n −+ − + − = − + − − −     (B.76) Substituting (B.74)-(B.76) into (B.39) and taking into account the notations (3.12)-(3.13), we obtain () 2 ,, , 2 ,, Q. 2 nn n u λ λ αρ β =+    ( B . 7 7 ) Theorem 6 is proved. Proof of Theorem 7 According to Remark 4, for 0 λ = , the function () ( ) 01 1 2 сс с θθ ϒ= + m a y b e u s e d instead of () λ θ ϒ . Since R const = , it follows from (B.52) that 0 =  . Thus, the transition (3.1) from the m omentum () , ρ θ to the coordinate () , x y representation takes the form () ( ) 0 ,, xy θ Φ= − Θ 1 sin , cos x c y θ θ ρ −    =       1 ,. 2 c r π φθ ρ == + (B.78) Let us simplify expressions (3.11) -(3.14) for the quantum pote ntial () () () 2 ,, 0 1 0 0 2 1 0, 0, 1 1 , 0, 0, 1 , 1 , 0, , , 1, 2 , k n k g Ag k g с c k =  −  −= =  −Θ = Θ   =    (B.79) () () ( ) 2 2 0 1 0 2 1 1, 0 , , 2 2 3 2 1 , 2 g с gn g n c Θ   −Θ = + − + − +     ( B . 8 0 ) () () ( ) 2 2 22 0 2 1 12 1 , 2 nn n c ρρ Θ  =+ − + + +      () () () ( ) () 2 2 2 11 2 1 0 22 2 2 0 21 1 Q, 1 , 0 , , , 24 nn nn n n TT nc c g с rr r αρ α ρθ ββ ρ ρ   ++ + =− Θ = − +   Θ         (B.81) which coincides with ( 3.21) when 1 T r c σ ρ = . We obtain the expression for the potential U from the Hamilton–Jacobi equation (3.16) 47 () () ( ) () 2 2 22 1 22 21 1 ,E , 4 nn rr nn n Ur c rr r σσ α φ β  ++ +  =− + − +       (B.82) which coincides with (3.22). Knowing the velocity 1 v с r α = , we obtain an expression for the probability density from formula (1.1) , n f  () ,, 11 exp , n n r n n n r fN nn r rr σσ  +   +     =−         ( B . 8 3 ) where the relations (1.11) are taken into account. The form of the wave function ψ follows directly from the expression fo r its phase (B.78) and the square of the modulus (B.83). Let us calculate the norm alisation integral () () () 2 1 4 23 1 00 3 ,, , 3 1 3 ,3 3 4 41 , 1 22 4 2 n r nn r n nk n n n nn n k k k kn kk r rr kn r n r nk k nn kk n dr fr d r N e N e d r N k nn n σ στ πσ πσ τ τ πσ σ + +∞ +∞ + − − + − + + + +− + + ++          ++     Γ        ==  = + =           () 2 22 2 , , 2 2 ,3 41 1 1 2 , 12 2 nn n nn k rr N fr d n rN kk nn n n n πσ πσ + ++ + −        Γ= Γ         =  =           (B.83) where, during integration, the variables 3 k =+  and 1 r τ = have been substituted. Expression (B.83) corresponds to (3. 20). Theorem 7 is proved. Calculation of the standard deviation Let us derive an expression for the s -th moment of the distribution () () () 2 1 2 1 0 4 , 3 0 3 2 ,, 3 , 1 3 , 1 2 2 2 4 41 1 4 2 4 n r n nn r n n nn ks n n n s r r s ks s k ks n n n n ks k r kn rr s r dr rf rd r N e r Ne d ks N N n ks n ks k nn n n ks n σ στ πσ πσ τ τ πσ σ πσ + +∞ − −− +∞ − ++ + ++ ++ − ++ ++ +    ++    ++ +     Γ=         ++  =  + == =  = ++ =             2 11 , s n k nn + +   Γ     () 2 2 2 ,, 2 , 12 2 1 n ss s n n r s rf rd N nn n r πσ + +    =Γ  +   −−        ( B . 8 4 ) 48 where the variables 3 ks =+ +  and 1 r τ = have been substituted. Using (B.84), we find r and 2 r , 32 4 34 , 11 ,, 13 14 22 nn n rr n rr nn n N nn N n πσ πσ       ΓΓ            −+  +− ==    (B.85) hence 2 2 22 2 2 2 12 , 4 2 13 n r r nn n n r n r σσ  −      −  ΓΓ − Γ       =   −          Γ  −  +− =     (B.86) where (B.83) is taken into account. Acknowledgements The study was conducted under a state comm ission from Lomonosov Moscow State University. References 1. H. Narnhofer, G.L. Sewell, Vlasov Hydrodynamics of a Quantum Me chanical Model, Communications in Mathematical Physics, 1981, vol. 79, pp. 9–24 2. V.I. Avrutskiy, O.I. Tolstikhin, Vlasov-equation approach to th e theory of dynam ic polarizability of multielectron systems: One-dimensional case, Physical Review A, 2024, vol. 110, No. 053106 3. S. Bastea, R. Esposito, J.L. Lebowitz, R. Marra, Hydrodynamics of Binary Fluid Phase Segregation, Physical Review Letters, 2002, vol. 89, No. 235701 4. M. Kopp, K. Vattis, C. Skordis, Solving the Vlasov equation in two spatial dimensions with the Schrödinger method, Physical Review D, 2017, vol. 96, no. 1 23532 5. S. Bergström, R. Catena, A. Chiappo, J. Conrad, B. Eurenius, M. Eriksson, M. Högberg, S. Larsson, E. Olsson, A. Unger, R. Wadman, J- factors for self-int eracting dark matter in 20 dwarf spheroidal galaxies, Phys ical Review D, 2018, vol. 98, no. 043017 6. L. Gabriel Gomez, J.A. Rueda, ‘Dark matter dynamical friction v ersus gravitational wave emission in the evolution of compact-star binaries’, Physical R eview D, 2017, vol. 96, no. 063001 7. D. Inman, Hao-Ran Yu, Hong-Ming Zhu, J.D. Emberson, Ue-Li Pen, Tong-Jie Zhang, Shuo Yuan, Xuelei Chen, Zhi-Zhong Xing, Simulating the cold dark mat ter-neutrino dipole with TianNu, Physical Review D, 2017, vol. 95, no. 083518 8. G . M a n f r e d i , J - L . R o u e t , B . M i l l e r , G . C h a r d i n , C o s m o l o g i c a l s t ructure form ation with negative mass, Physical Review D, 2018, vol. 98, No. 023514 9. J. Veltmaat, J. C. Niemeyer, B. Schwabe, Formation and structur e of ultralight bosonic dark matter halos, Physical Review D, 2018, vol. 98, No. 043509 10. D.V. Shetty, A.S. Botvina, S.J. Yennello, G.A. Souliotis, E. Be ll, A. Keksis, Fragment yield distribution and the influence of neutron composition and excitation energy in multifragmentation reactions, Physical Review C, 2005, vol. 71( 2). 11. H. Zheng, S. Burrello, M. Colonna, D. Lacroix, G. Scamps, Conne cting the nuclear equation of state to the interplay between fusion and quasifission proce sses in low-energy nuclear reactions. Physical Review C, 2018, vol. 98(2). 49 12. D. Pierroutsakou, B. Martin, C. Agodi, R. Alba, V. Baran, A. Bo iano, … C. Signorini, Dynamical dipole mode in fusion reactions at 16 MeV/nucleon and b e a m e n e r g y dependence. Physical Review C, 2009, vol. 80(2). 13. B. Atenas, S. Curilef, Dynamics and thermodynamics of systems with l ong-range dipole-type interactions, Physical Review E, 2017, vol. 95, no. 022110 14. M. Giona, Space-time-modulated sto chastic processes, Physical R eview E, 2017, vol. 96, No. 042132 15. P. Kumar, B.N. Miller, ‘Thermodynamics of a one-dimensional self-gravitating gas with periodic boundary conditions’, P hysical Review E, 2017, vol. 95, no. 022116 16. K. Hara, I. Barth, E. Kaminski, I.Y. Dodin, N.J. Fisch, Kinetic simulations of ladder climbing by electron plasma waves, Physical Review E, 2017, vol . 95, no. 053212 17. M. Horky, W .J. Miloch, V.A. Delong, Numerical heating of electr ons in par ticle-in-cell simulations of fully magnetised plas mas, Physical Review E , 201 7, vol. 95, no. 043302 18. N . R a t a n , N . J . S i r c o m b e , L . C e u r v o r s t , J . S a d l e r , M . F . K a s i m , J . Holloway, M.C. Levy, R. Trines, R. Bingham, P.A. Norreys, Dense plasma heating by cross ing relativistic electron beams, Physical Review E, 2017, vol. 95, no. 013211 19. L.D. Landau, E.M. Lifshitz, Quant um Mechanics, vol. 3, Pergamon Press, 1977, 677 p. 20. Vlasov A.A., Statistical Distrib ution Functions. Nauka, Moscow, 1966 21. Vlasov A.A. , Many-Particle Theory and Its Application to Plasma . – New York: Gordon and Breach, 1961. 22. Perepelkin E.E., Sadovnikov B.I., Inozemtseva N.G., Aleksandrov I.I., Dispersion chain of Vlasov equations //J. Stat. Mech. – 2022. – No. 013205. 23. M. Abraham, Theory of Electricity. Volume II: Electromagnetic T heory of Radiation, Monatshefte für Mathematik und Physik, 1906, vol. 17 (1): A39. ISSN 0026-9255. 24. A.O. Barut, Electrodynamics and classical theory of fields & pa rticles. New York: Dover Publications, 1980, pp. 179–184. ISBN 0-486-64038-8. 25. E.E. Perepelkin, B.I. Sadovnikov, N.G. Inozemtseva, The propert ies of the first equation of the Vlasov chain of equations //J. Stat. Mech. – 2015. – No. P0 5019. 26. E.E. Perepelkin, B.I. Sadovnikov, N.G. Inozemtseva, M.V. Kli menko, Construction of Schrödinger, Pauli and Dirac equations from the Vlasov equation in the case of t he Lorentz gauge, European Physical Journal Plus, 2025, vol. 140, No. 622, pp. 1–23 27. D. Bohm, B.J. Hiley, Kaloyerou, P.N., An ontological basis for the quantum theory. Phys. Rep., 1987, vol. 144, pp. 321–375. 28. D. Bohm, B.J. Hiley, The Undivi ded Universe: An Ontological Int erpretation of Quantum Theory, 1993, Routledge, London. 29. L. de Broglie, A Causal and Non-linear Interpretation of Wave Mechanics: The Theory of the Double Solution, 1956, G authiers-Villiars, Paris. 30. E.E. Perepelkin, B.I. Sadovnikov, N.G. Inozemtseva, E.V. Burlak ov, P.V. Afonin, Is the Moyal equation for the Wigner function a quantum analogue of th e Liouville equation?, //Journal of Statistical Mechan ics, 2023, No. 093102, pp. 1–21 31. E. Moyal, Quantum mechanics as a statistical t heory //Proceedin gs of the Cambridge Philosophical Society, 1949, vol. 45, pp. 99–124. 32. Wigner E.P. On the quantum correction for thermodynamic equilibrium //Phys. Rev. – 1932. – vol. 40 – pp. 749–759. 33. H. Weyl, The theory of groups a nd quantum mechanics (Dover, New York, 1931). 34. R. Courant, D. Hilbert, Methods of Mathematical Physics. Vol. I I: Partial Differential Equations, Interscience, New York, London, 1962 35. M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equ ations and Inverse Scattering, Cambridge Unive rsity Press, Cambridge, 1991 36. M.D. Kruskal, A. Ramani, B. Gra mmaticos, Singularity analysis and its relation to complete, partial and non-integrability. In: R. Conte, N. Boccara (eds.) Partially integrable evolution equations in physics, Kluwer Academic Publishers, Dordrecht, 1990. 50 37. N.K. Ibragimov, Group analysis of ordinary differential equations and the invariance principle in mathematical physic s (for the 150th anniversary of Sophus Lie). Russian Math Surv 47(4), 89, 1992. 38. P. Painlevé, Lectures on the Anal ytic Theory of Differential Eq uations, delivered in Stockholm, Paris, 1895 39. P. Painlevé, Mémoir sur les équations différentielles dont l’in tég rale générale est uniforme, Bull. Soc. Math. France 28, 201–261 (1900) 40. P. Painlevé, On second-order a nd higher-order differential equa tions whose general integrals are uniform. Acta Mathematica, 1902, vol. 25, pp. 1–85 41. V.E. Adler, A.B. Shabat, R.I. Yamilov, A symmetry approach to t he integrability problem. Theoretical and Mathematical Physics, 2000, vol. 125, pp. 1603– 1661. 42. M. Musette, Painlevé analysis for nonlinear partial differentia l equations. In: Conte, R. (ed.) The Painlevé Property, One Century Later. CRM series in Mathema tical Physics, pp. 517– 572. Springer, New York 1999. 43. P.D. Lax, Integrals of nonlinear equations of evolution and solitary waves. Communications on Pure and Applied Mathematics, 1968, vol. 21, pp. 467–490. 44. The ATLAS Collaboration, G. Aad, E. Abat, J. Abdallah, A.A. Abd elalim, A. Abdesselam, O. Abdinov, B.A. Abi, M. Abolins, H. Abramowicz, The A TLAS Expe riment at the CERN Large Hadron Collider, Journal of Instrum entation, 2008, vol. 3 , No. S08003 45. B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, C . A d a m s , T . A d a m s , P . Addesso, R. X. Adhikari et al. (L IGO Scientific Collabo ration a nd Virgo Collaboration), GW170814: A Three-Detector Observation of Gravitational Waves f rom a Binary Black Hole Coalescence, Phys. Rev. Lett., 2017, vol. 119, No. 141101 46. A. Quarteroni, R. Sacco, F. Saleru, Numerical Mathematics, Spri nger, 1991, ISBN 0-387- 98959-5 47. P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman Pub lishing Inc, New York, 1985, p. 422, ISBN 0-273-08647-2 48. E.E. Perepelkin, B.I. Sadovnikov, N.G. Inozemtseva, Solutions o f nonlinear equations of divergence type in domains havi ng corner points, Journal of Ell iptic and Parabolic Equations, 2018, vol. 4, no. 1, pp. 107–139 49. T. Shiroto, N. Ohnishi, and Y. Sentoku, ‘Quadratic conservative sche me f or relativis tic Vlasov-Maxwell system’, Journal of Com putational Physics, 2019, vol. 379, pp. 32–50. 50. Druyvesteyn, M. 1930, Physica, 10, 61 51. Tischmann, S., Gaelzer, R., Schröder, D.L., Lazar, M., & Fichtn er, H. (2025). Electrostatic Waves in Astrophysical Druyvesteyn Plasmas. I. Langmuir Waves. The Astrophysical Journal, 997. 52. Takahashi, K., Charles, C., Boswell, R.W., & Fujiwara, T. (2011 ). Electron energy distribution of a current-free double layer: Druyvesteyn theory and experiments. Physical Review Letters, 107(3), 035002 53. Bakiev, T.N., Nakashidze, D.V., Savchenko, A.M., & Semenov, K.M . (2022). Generalised Maxwell Distribution in the Tsallis Entropy Formalism. Moscow U niversity Physics Bulletin, 77, 728–740. 54. Shama, M.S., Alharthi, A.S., Almulhim, F.A., Gemeay, A.M., Mera ou, M.A., Mustafa, M.S., Hussam, E., & Aljohani, H.M. (2023). Modified generalised Weibu ll distribution: theory and applications. Scientific Reports, 13. 51 55. Noori, N.A., Abdullah, K.N., & Khaleel, M.A. (2025). Developmen t and Applications of a New Hybrid Weibull-Inverse Wei bull Distribution. Modern Journal of Statis tics. 56. Afungchui, D., & Aban, C.E. (2023). Analysis of wind regimes fo r energy estimation in Bamenda, in the North-West Region of Cameroon, based on the Wei bull distribution. J ournal of Renewable Energies. 57. Klakattawi, H.S. (2022). Survival analysis of cancer patients u sing a new extended Weibull distribution. PLoS ONE, 17. 58. M. Arnaud, J. Bardoux , F. Bergsma , G. Bobbink , A. Bruni , L. C h e v a l i e r , P . E n n e s , P . Fleischmann , M. Fontaine , A. Formica , V. Gautard, H. Groenst ege, C. Guyot, R. Hart, W. Kozanecki , P. Iengo , M. Legendre , T. Nikitina , E. Perepelki n , P. Ponsot…S. Vorozthsov, Commissioning of the magnetic fi eld in t he A TL A S mu o n s pe c tr om e te r, Nu cl e ar Ph y si cs B - Proceedings Supplements, 2008, vol. 177–178, pp 265–266 59. A. Abada, M. Abbrescia, S.S. Abdus Salam et al. FCC Physics Opp ortunities, The European Physical Journal C, 2019, vol. 79, no. 474 60. A. Abada, M. Abbrescia, S.S. Abdus Salam et al., HE-LHC: The Hi gh-Energy Large Hadron Collider, The European Physical Journal Special Topics, 2019, v ol. 228, pp. 1109–1382 61. Perepelkin E.E., Sadovnikov B.I., Inozemtseva N.G., Ψ-model of micro- and macrosystems, Annals of Physics, 383 (2017) 511–544 62. Zhidkov E.P., Perepelkin E.E. An analytical approach for quasi- linear equations of t he second order. SMAM, vol. 1 (2001), No. 3, pp. 285–297. 63. R. Courant and D. Hilbert, Methods of Mathematical Physics. Par tial Differential Equations. Vol. 2, 1962, New York, London. 64. E.E. Perepelkin, B.I. Sadovnikov, N.G. Inozemtseva, P. V. Afonin , Wigner function for electromagnetic systems, Physical Review A, 2024, vol. 110, No. 02222, pp. 1–24