Quantum integrable systems in three-dimensional magnetic fields: the Cartesian case

Quan tum in tegrable syst e ms in three- d imensional magnetic fields : the Cartes ian case Alexander Zhalij Institute o f Ma thematics of NAS of Ukraine, 3 T ereshc henkivsk a Str., 01601 Kyiv-4, Ukraine zhaliy@imath.kiev.ua In this pap er we construct in tegra ble three-dimens io nal quan tum-mechanical systems with mag- netic fields, admitting pairs of comm uting second-or der int egr als of motion. The cas e of Carte- sian co ordina tes is considered. Most of the systems obtained are new and not rela ted to the separatio n of v ariables in the cor resp onding Schr¨ odinger eq uation. Let u s consider the s tationary Sc hr ¨ odinger equation for particle mo ving in external electromag- netic field in three-dimensional Euclidean space H ψ = E ψ, H = 1 2 ~ p 2 + V ( x, y , z ) + A i ( x, y , z ) p i + p i A i ( x, y , z ) , (1) where V ( x, y , z ) and ~ A = ( A 1 ( x, y , z ) , A 2 ( x, y , z ) , A 3 ( x, y , z )) are scalar and v ector p oten tials of electromagnetic field, resp ectiv ely . Here and b elo w we use the notation ~ p = − ih ~ ∇ and the summation from 1 to 3 o v er the r ep eated in dices is understo o d . By analogy with classic al Hamiltonian mec hanics this system is ca lled integrable if there exists a pair of qu antum-mec hanical op erators P and Q which commute with eac h other as well as with the Hamiltonian H , i.e., the follo wing r elations hold [ H , Q ] = [ H , P ] = [ P , Q ] = 0 . Moreo v er, all thr ee op er ators H , Q and P are algebraically indep en den t, i.e., an y of them cannot b e represente d as p olynomial of tw o others [1]. In this pap er w e restrict ourselves by th e case of Q and P b eing quadr atic p olynomials of ~ p Q = α ik ( x, y , z ) p i p k + f i ( x, y , z ) p i + γ 1 ( x, y , z ) , P = β ik ( x, y , z ) p i p k + g i ( x, y , z ) p i + γ 2 ( x, y , z ) , (2) In classical m echanics in tegrable systems are interesting since their motion in a phase space is m ore ordered, namely , it is restricted to a toru s. In qu an tum mec hanics in tegrabilit y of an n -dimensional quan tum-mechanica l system, i.e., existing of n quantum inte grals of motion (op- erators, whic h commute w ith eac h other as we ll as with the op erator of equation (1)), sim p lifies the problem of d etermin in g of energy sp ectrum and wa v e fun ctions ev en w h en in tegrabilit y do es not lead to the v ariables separation, b ut only to the so-called “quasi-separation of v ariables” [2]. Therefore, th e classification problem of all p oten tials V and ~ A of electromag netic field, for whic h quant u m -mec hanical pr ob lem is in tegrable in the sense describ ed ab o ve , is of curren t imp ortance. In the three-dimensional case of s calar p otent ial, when there is n o any m agnetic field, this problem was solv ed as early as in 1967 by Y ak o v A. Smoro dinsky w ith coauthors [3, 4]. Th ey ha ve pro ved that if there exist t w o quantum inte grals of motion of fi rst or second order with resp ect to ~ p (in the sense describ ed ab o ve ) then there exists a p ossibilit y of v ariable separation 1 in the corresp onding Schr¨ odinger equation [4]. The in verse s tatemen t is also tru e [3]. Th us, elev en classes of inte grable p oten tials V obtained by them coincide w ith the results of classic pap er of Eisenh art [5], wh er e h e as early as in 1948 has describ ed all scalar p oten tials for whic h corresp onding Sc h r ¨ odinger equation (or Hamilton-Jacobi equation in classical mec han ics) admits v ariable separation at least in one of elev en coord inate systems. The next step w as done in 19 72 by V.N. S hap o v alo v with coauthors [6]. They ha v e ob- tained co mp lete cla ssification list of vec tor-p oten tials with n on zero magnetic field, for w h ic h corresp onding Schr¨ odinger equation (1) admits v ariable separation at least in one of eleve n co- ordinate systems, and list of corresp onding pairs of op erators wh ic h commute w ith eac h other and with op erator of the equation. P osterior r esults in this direction are connected with wo rks of P . Winternitz and his coau- thors [7–9]. F or t w o-dimensional case they hav e disco v ered that existing of second-order in tegrals of motion in magnetic field do es not guaran tee p ossibility of v ariable sep aration. Nev ertheless, ev en in this case, in tegrals of motion are classified on equiv alence classes u nder action of Euclid- ian group an d second-order terms with r esp ect to p i in these in tegrals h a v e the same form as in th e case of pu re scalar p otentia l. It w as also shown that in m agnetic field qu an tum case [9] do es not obvi ously coincide with classical one [7, 8], namely , constructed ve ctor-p otent ials can dep end on Plank constan t ~ in a nontrivial wa y . In this pap er w e make the next step in classification of p oten tials of electromagnetic field V and ~ A in three-dimensional Euclidian space, f or whic h corresp onding quantum-mec hanical sys- tem describ ed b y equation (1 ) is in tegrable in th e sens e explained ab o v e, i.e., for w hic h there exists a pair of op erators (2) that comm u te with eac h other as well as with the op erator of the equation. As a result, we obtain a num b er of ve ctor-p otent ials for whic h corresp onding Sc hr¨ odin ger equation is in tegrable bu t do es n ot admit v ariable sep aration. Therefore, these p oten tials d id not app ear in the Sh ap o v alo v’s classification [6] and are new. A t fi rst w e consid er single op erator Q of the form (2) whic h commute s with the op erator H of S c hr¨ odinger equation (1) . The comm utator [ Q, H ] cont ains terms of zero, first, second and third orders with resp ect to p i , whic h co efficien ts hav e to v anish. Co efficient s of the third p o w er of p i giv e the follo wing system: ∂ α ik ∂ x m p m p i p k = 0 . After solving this sys tem we obtain th at the op erator Q can b e presente d as symmetric bilinear p olynomial of th e infinitesimal generators of a group of m otions of th ree-dimensional Euclidean space E 3 , i.e., symmetry group of the three-dimensional Sc h r ¨ odinger equation for free particle (the Helmholtz equation): Q = a ik M i M k + b ik ( p i M k + M k p i ) + c ij p i p k + f i ( x, y , z ) p i + γ 1 ( x, y , z ) , (3) where a ik , b ik and c ij are constan ts, M i is op erator of rotation, n amely , M i = ε ik l x k p l , where ε ik l is completely an tisymmetric tensor. Comparing constructed form (3) of op erator with similar form of op erator in the case of pure scalar p oten tial [4], we conclude that second-order terms with resp ect to p i remain the same after app earing of nonzero magnetic field (as w ell as in tw o-dimensional case [9]). Therefore, analogously to [4], a pair of comm uting op erators P and Q ha ving the form (3 ) can b e r educed b y rotations and trans lations of co ordinate system an d b y the transformations Q ′ = µP + ν Q + λH to one of elev en classes corresp onding to elev en classical co ord inate systems which pr o vide separation of v ariable in the three-dimensional Sc hr¨ odinger equation for free particle. 2 That is why in our case of nonzero m agnetic fi eld w e obtain that the second-order term s with resp ect to p i in these elev en comm uting pairs of the op erators P and Q are of the same form as ones pr esen ted in [4]. Ho w eve r, in con trast to the pu re scalar case, some or all of the co efficients of the first p o w er of p i are nonzero functions. The f orms of these functions d etermine the form of magnetic fi eld (in scalar case they equal zero). In this pap er w e completely solve the s im p lest “Cartesian” case, i.e. th e case Q = p 2 1 + ~ f ( x, y , z ) ~ p + γ 1 ( x, y , z ) , (4) P = p 2 2 + ~ g ( x, y , z ) ~ p + γ 2 ( x, y , z ) . (5) Splitting th e equations [ H , Q ] = [ H , P ] = [ P , Q ] = 0 with r esp ect to d ifferen t p o wers of p i , w e obtain an o v erdeterminin g system of PDEs on un kno wn fun ctions f 1 , f 2 , f 3 , g 1 , g 2 , g 3 , γ 1 , γ 2 and V , A 1 , A 2 , A 3 . The co efficien ts of th e highest p o w ers of p i giv e the follo wing system f 2 = f 2 ( x ) , f 3 = f 3 ( x ) , g 1 = g 1 ( y ) , g 3 = g 3 ( y ) , f 1 y = g 2 x , f ′ 2 ( x ) + f 1 y = 4 A 2 x , g ′ 3 ( y ) + g 2 z = 4 A 3 y , 4 A 1 x = f 1 x , f ′ 3 ( x ) + f 1 z = 4 A 3 x , g ′ 1 ( y ) + g 2 x = 4 A 1 y , 4 A 2 y = g 2 y . Its general solution for ~ A is 4 A 1 = s x + k 1 x + g 1 ( y ) + r 1 ( z ) , 4 A 2 = s y + k 2 y + f 2 ( x ) + r 2 ( z ) , 4 A 3 = s z + k 1 z + k 2 z + f 3 ( x ) + g 3 ( y ) + r ′ 3 ( z ) , where s = s ( x, y , z ), k 1 = k 1 ( x, z ) and k 2 = k 2 ( y , z ). The gauge transformation ~ A → ~ A + ~ ∇ F , F = s ( x, y , z ) + k 1 ( x, z ) + k 2 ( y , z ) + r 3 ( z ) , simplifies the obtained expression for ~ A in the follo wing w a y A 1 = 1 4 ( g 1 ( y ) + r 1 ( z )) , A 2 = 1 4 ( f 2 ( x ) + r 2 ( z )) , A 3 = 1 4 ( f 3 ( x ) + g 3 ( y )) . With this exp ression for ~ A in hand, we obtain from th e co efficien ts of the lo west p o wers of p i the system of ODEs on the functions g 1 ( y ), r 1 ( z ) f 2 ( x ), r 2 ( z ), f 3 ( x ) and g 3 ( y ): f 2 ( x ) g ′ 3 ( y ) = g 1 ( y ) f ′ 3 ( x ) , r 1 ( z ) f ′ 2 ( x ) = f 3 ( x ) r ′ 2 ( z ) , (6) r 2 ( z ) g ′ 1 ( y ) = g 3 ( y ) r ′ 1 ( z ) . It is obvious that the Sc hr¨ odin ger equation with vec tor-p oten tial (1) is inv a riant with resp ect to p erm utations of A 1 , A 2 and A 3 , which are don e sim ultaneously with p ermutations of the v ariables x 1 , x 2 and x 3 . Equations (6) are in v arian t with resp ect to p erm utations of th e fun ctions g 1 ( y ), r 1 ( z ), f 2 ( x ), r 2 ( z ), f 3 ( x ) and g 3 ( y ). Th ese equiv alence transformations can b e represente d 3 in suc h a wa y     f 2 g 3 r 1 f 3 g 1 r 2 A 1 A 2 A 3 x y z     ∼     r 1 f 2 g 3 r 2 f 3 g 1 A 3 A 1 A 2 z x y     ∼     g 3 r 1 f 2 g 1 r 2 f 3 A 2 A 3 A 1 y z x     ∼     f 3 r 2 g 1 f 2 r 1 g 3 A 1 A 3 A 2 x z y     ∼     g 1 f 3 r 2 g 3 f 2 r 1 A 2 A 1 A 3 y x z     ∼     r 2 g 1 f 3 r 1 g 3 f 2 A 3 A 2 A 1 z y x     . Usage of these equiv alence transform ations allo ws us to describ e exhaustively all solutions of equations (6). This giv es the complete d escription of all p ossible forms of vecto r-p oten tials ~ A . The residual co efficien ts of p o w ers of p i serv e for determining of scalar comp onent V of v ector- p oten tial and put additional constrain ts on the fun ctions g 1 ( y ), r 1 ( z ), f 2 ( x ), r 2 ( z ), f 3 ( x ) and g 3 ( y ). Belo w we adduce the final r esults of our calculations. Here and b elo w ~ Ω denotes magnetic field, namely , ~ Ω = rot ~ A . Case 1. ~ A = 0 , ~ Ω = 0 , V = u 1 ( x ) + u 2 ( y ) + u 3 ( z ) , Q = p 2 1 + 2 u 1 ( x ) , P = p 2 2 + 2 u 2 ( y ) . This case corresp onds to zero magnetic field and is con tained in the Eisenhart’s classification [5]. According to his results all scalar p oten tials for whic h corresp onding Sc hr¨ odinger equations admit v ariable separation in Cartesian co ordin ates are exhau s ted by ones h a ving the form V = u 1 ( x ) + u 2 ( y ) + u 3 ( z ) . Case 2. ~ A =   v 1 ( z ) v 2 ( z ) 0   , ~ Ω =   − v ′ 2 ( z ) v ′ 1 ( z ) 0   , V = v 3 ( z ) , Q = p 2 1 , P = p 2 2 . Case 3. ~ A =   0 0 f ( x ) + g ( y )   , ~ Ω =   g ′ ( y ) − f ′ ( x ) 0   , V = u 1 ( x ) + u 2 ( y ) , Q = p 2 1 + 4 f ( x ) p 3 + 2 u 1 ( x ) , P = p 2 2 + 4 g ( y ) p 3 + 2 u 2 ( y ) . The cases 2 an d 3 where obtained b y Shap o v a lo v et al. [6]. According to their results these t w o cases exhaust all v ector-p oten tials with nonzero m agnetic field for whic h corresp onding Sc hr¨ odin ger equations admit v ariable separation in Cartesian co ordinates. Next thr ee cases are not connected with v ariable s eparation and , therefore, th ese p otentia ls did not app ear in the Shap o v a lov’ s classificatio n [6] and they are new. 4 Case 4. ~ A =   g ′ ( y ) f ′ ( x ) 0   , ~ Ω =   0 0 f ′′ ( x ) − g ′′ ( y )   , V = − ( C 3 f ( x ) + C 3 g ( y ) + 2 C 2 f 2 ( x ) + 2 C 1 g 2 ( y ) + r ( z ) + 4 g ( y ) f ′′ ( x ) + 4 f ( x ) g ′′ ( y )) , Q = p 2 1 + 4 f ′ ( x ) p 2 − 2(4 g ( y ) f ′′ ( x ) + 2 C 2 f ( x ) 2 + C 3 f ( x )) , P = p 2 2 + 4 g ′ ( y ) p 1 − 2(4 f ( x ) g ′′ ( y ) + 2 C 1 g ( y ) 2 + C 3 g ( y )) , where the f unctions f ( x ) and g ( y ) are solutions of the ODEs f ′′ ( x ) = C f 2 ( x ) + C 1 f ( x ) + C 4 , (7) g ′′ ( y ) = C g 2 ( y ) + C 2 g ( y ) + C 5 . (8) If C = 0 these are linear second-ord er O DEs. The case C 6 = 0 is more in teresting. If additionally C 1 6 = 0 (resp . C 2 6 = 0), then solution of equation (7) (resp. (8)) is fir st Pai n leve transcendent. If C = C 1 = 0 (resp . C = C 2 = 0), then (7) (resp . (8)) is the W eierstrass equation wh ic h s olutions are expressed either via W eierstrass functions or v ia element ary ones dep end ing on v alues of the parameter C 4 (resp. C 5 ) and in tegration constants. S ee, e.g. [10] for details. Case 5. ~ A =   g ′ ( y ) f ′ ( x ) C f ( x ) + C g ( y )   , ~ Ω =   C g ′ ( y ) − C f ′ ( x ) f ′′ ( x ) − g ′′ ( y )   , V = − ( C 3 f ( x ) + C 3 g ( y ) + 2 C 2 f 2 ( x ) + 2 C 1 g 2 ( y ) + 4 g ( y ) f ′′ ( x ) + 4 f ( x ) g ′′ ( y )) , Q = p 2 1 + 4( f ′ ( x ) p 2 + C f ( x ) p 3 ) − 2(4 g ( y ) f ′′ ( x ) + 2 C 2 f ( x ) 2 + C 3 f ( x )) , P = p 2 2 + 4( g ′ ( y ) p 1 + C g ( y ) p 3 ) − 2(4 f ( x ) g ′′ ( y ) + 2 C 1 g ( y ) 2 + C 3 g ( y )) , where the f unctions f ( x ) and g ( y ) are solutions of the ordinary differen tial equations f ′′ ( x ) = C 6 f 2 ( x ) + C 1 f ( x ) + C 4 , g ′′ ( y ) = C 6 g 2 ( y ) + C 2 g ( y ) + C 5 , whic h are integ rated analogously to the previous case. Case 6. ~ A = 1 4   w ′ 1 ( y ) + v ′ 1 ( z ) u ′ 2 ( x ) + v ′ 2 ( z ) u ′ 3 ( x ) + w ′ 3 ( y )   , ~ Ω = 1 4   w ′′ 3 ( y ) − v ′′ 2 ( z ) v ′′ 1 ( z ) − u ′′ 3 ( x ) u ′′ 2 ( x ) − w ′′ 1 ( y )   , V = − 1 4 ( u 1 ( x ) + w 2 ( y ) + v 3 ( z ) + w 1 ( y ) u ′′ 2 ( x ) + v 1 ( z ) u ′′ 3 ( x ) + u 2 ( x ) w ′′ 1 ( y ) + v 2 ( z ) w ′′ 3 ( y ) + u 3 ( x ) v ′′ 1 ( z ) + w 3 ( y ) v ′′ 2 ( z )) , Q = p 2 1 + u ′ 2 ( x ) p 2 + u ′ 3 ( x ) p 3 − 1 2 ( w 1 ( y ) u ′′ 2 ( x ) + v 1 ( z ) u ′′ 3 ( x ) + u 1 ( x )) , P = p 2 2 + w ′ 1 ( y ) p 1 + w ′ 3 ( y ) p 3 − 1 2 ( u 2 ( x ) w ′′ 1 ( y ) + v 2 ( z ) w ′′ 3 ( y ) + w 2 ( y )) , 5 where the f unctions u 2 ( x ), u 3 ( x ), w 1 ( y ), w 3 ( y ), v 1 ( z ), v 2 ( z ), u 1 ( x ), w 2 ( y ) and v 3 ( z ) are defined in a sp ecial w a y and describ ed b y the follo wing four cases: Case 6.1. u 2 ( x ) = a 3 ( r 1 cosh( a 1 x ) + k 1 sinh( a 1 x )) , u 3 ( x ) = a 2 ( r 1 sinh( a 1 x ) + k 1 cosh( a 1 x )) , w 1 ( y ) = a 3 ( r 2 cosh( a 2 y ) + k 2 sinh( a 2 y )) , w 3 ( y ) = a 1 ( r 2 sinh( a 2 y ) + k 2 cosh( a 2 y )) , v 1 ( z ) = a 2 ( r 3 cosh( a 3 z ) + k 3 sinh( a 3 z )) , v 2 ( z ) = a 1 ( r 3 sinh( a 3 z ) + k 3 cosh( a 3 z )) , u 1 ( x ) = a 2 2 a 2 3 4  ( r 2 1 + k 2 1 ) cosh(2 a 1 x ) + 2 r 1 k 1 sinh(2 a 1 x )  + C ( r 1 cosh( a 1 x ) + k 1 sinh( a 1 x )) , w 2 ( y ) = a 2 1 a 2 3 4  ( r 2 2 + k 2 2 ) cosh(2 a 2 y ) + 2 r 2 k 2 sinh(2 a 2 y )  + C ( r 2 cosh( a 2 y ) + k 2 sinh( a 2 y )) , v 3 ( z ) = a 2 1 a 2 2 4  ( r 2 3 + k 2 3 ) cosh(2 a 3 z ) + 2 r 3 k 3 sinh(2 a 3 z )  + C 1 ( r 3 cosh( a 3 z ) + k 3 sinh( a 3 z )) , with 5 p ossible sub cases: a) C = 0 , C 1 = 0; b) r 1 = k 1 , r 2 = k 2 , r 3 = k 3 , C 1 = C ; c) r 1 = k 1 , r 2 = − k 2 , r 3 = − k 3 , C 1 = C ; d) r 1 = − k 1 , r 2 = k 2 , r 3 = − k 3 , C 1 = − C ; e) r 1 = − k 1 , r 2 = − k 2 , r 3 = k 3 , C 1 = − C. Case 6.2. u 2 ( x ) = a 3 ( r 1 sin( a 1 x ) − k 1 cos( a 1 x )) , u 3 ( x ) = a 2 ( r 1 cos( a 1 x ) + k 1 sin( a 1 x )) , w 1 ( y ) = a 3 ( r 2 sin( a 2 y ) − k 2 cos( a 2 y )) , w 3 ( y ) = a 1 ( r 2 cos( a 2 y ) + k 2 sin( a 2 y )) , v 1 ( z ) = a 2 ( r 3 cosh( a 3 z ) + k 3 sinh( a 3 z )) , v 2 ( z ) = a 1 ( r 3 sinh( a 3 z ) + k 3 cosh( a 3 z )) , u 1 ( x ) = a 2 2 a 2 3 4  ( r 2 1 − k 2 1 ) cos(2 a 1 x ) + 2 r 1 k 1 sin(2 a 1 x )  + C ( r 1 sin( a 1 x ) − k 1 cos( a 1 x )) , w 2 ( y ) = a 2 1 a 2 3 4  ( r 2 2 − k 2 2 ) cos(2 a 2 y ) + 2 r 2 k 2 sin(2 a 2 y )  + C ( r 2 sin( a 2 y ) − k 2 cos( a 2 y )) , v 3 ( z ) = − a 2 1 a 2 2 4  ( r 2 3 + k 2 3 ) cosh(2 a 3 z ) + 2 r 3 k 3 sinh(2 a 3 z )  + C 1 ( r 3 cosh( a 3 z ) + k 3 sinh( a 3 z )) , with 5 p ossible sub cases: a) C = 0 , C 1 = 0; b) r 1 = ik 1 , r 2 = − ik 2 , r 3 = − k 3 , C 1 = iC ; c) r 1 = ik 1 , r 2 = ik 2 , r 3 = k 3 , C 1 = iC ; d) r 1 = − ik 1 , r 2 = − ik 2 , r 3 = k 3 , C 1 = − iC ; e) r 1 = − ik 1 , r 2 = ik 2 , r 3 = − k 3 , C 1 = − iC. 6 Case 6.3. u 2 ( x ) = a 3 ( r 1 cos( a 1 x ) + k 1 sin( a 1 x )) , u 3 ( x ) = ia 2 ( r 1 sin( a 1 x ) − k 1 cos( a 1 x )) , w 1 ( y ) = a 3 ( r 2 cos( a 2 y ) + k 2 sin( a 2 y )) , w 3 ( y ) = ia 1 ( r 2 sin( a 2 y ) − k 2 cos( a 2 y )) , v 1 ( z ) = a 2 ( r 3 cos( a 3 z ) + k 3 sin( a 3 z )) , v 2 ( z ) = ia 1 ( r 3 sin( a 3 z ) − k 3 cos( a 3 z )) , u 1 ( x ) = − a 2 2 a 2 3 4  ( r 2 1 − k 2 1 ) cos(2 a 1 x ) + 2 r 1 k 1 sin(2 a 1 x )  + C ( r 1 cos( a 1 x ) + k 1 sin( a 1 x )) , w 2 ( y ) = − a 2 1 a 2 3 4  ( r 2 2 − k 2 2 ) cos(2 a 2 y ) + 2 r 2 k 2 sin(2 a 2 y )  + C ( r 2 cos( a 2 y ) + k 2 sin( a 2 y )) , v 3 ( z ) = − a 2 1 a 2 2 4  ( r 2 3 − k 2 3 ) cos(2 a 3 z ) + 2 r 3 k 3 sin(2 a 3 z )  + C 1 ( r 3 sin( a 3 z ) − k 3 cos( a 3 z )) , with 5 p ossible sub cases: a) C = 0 , C 1 = 0; b) r 1 = ik 1 , r 2 = − ik 2 , r 3 = ik 3 , C 1 = iC ; c) r 1 = − ik 1 , r 2 = − ik 2 , r 3 = − ik 3 , C 1 = iC ; d) r 1 = − ik 1 , r 2 = ik 2 , r 3 = ik 3 , C 1 = − iC ; e) r 1 = ik 1 , r 2 = ik 2 , r 3 = − ik 3 , C 1 = − iC. Case 6.4. u 2 ( x ) = a 3 ( r 1 cosh( a 1 x ) + k 1 sinh( a 1 x )) , u 3 ( x ) = − ia 2 ( r 1 sinh( a 1 x ) + k 1 cosh( a 1 x )) , w 1 ( y ) = a 3 ( r 2 cosh( a 2 y ) + k 2 sinh( a 2 y )) , w 3 ( y ) = − ia 1 ( r 2 sinh( a 2 y ) + k 2 cosh( a 2 y )) , v 1 ( z ) = a 2 ( r 3 cos( a 3 z ) + k 3 sin( a 3 z )) , v 2 ( z ) = ia 1 ( r 3 sin( a 3 z ) − k 3 cos( a 3 z )) , u 1 ( x ) = a 2 2 a 2 3 4  ( r 2 1 + k 2 1 ) cosh(2 a 1 x ) + 2 r 1 k 1 sinh(2 a 1 x )  + C ( r 1 cosh( a 1 x ) + k 1 sinh( a 1 x )) , w 2 ( y ) = a 2 1 a 2 3 4  ( r 2 2 + k 2 2 ) cosh(2 a 2 y ) + 2 r 2 k 2 sinh(2 a 2 y )  + C ( r 2 cosh( a 2 y ) + k 2 sinh( a 2 y )) , v 3 ( z ) = a 2 1 a 2 2 4  ( r 2 3 − k 2 3 ) cos(2 a 3 z ) + 2 r 3 k 3 sin(2 a 3 z )  + C 1 ( r 3 sin( a 3 z ) − k 3 cos( a 3 z )) , with 5 p ossible sub cases: a) C = 0 , C 1 = 0; b) r 1 = − k 1 , r 2 = − k 2 , r 3 = − ik 3 , C 1 = C ; c) r 1 = k 1 , r 2 = − k 2 , r 3 = ik 3 , C 1 = C ; d) r 1 = k 1 , r 2 = k 2 , r 3 = − ik 3 , C 1 = − C ; e) r 1 = − k 1 , r 2 = k 2 , r 3 = ik 3 , C 1 = − C. 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