On the Applications of a New Technique to Solve Linear Differential Equations, with and without Source

Symmetry , Integrabilit y and Geometry: Metho ds and Applications SIGMA 3 (2007), 057, 6 pages On the Application s of a Ne w T ec hnique to S olv e Line ar Dif feren tial Equations , with and w ithout Source ⋆ N. GURAPP A † , Pankaj K. JHA ‡ and Pr asanta K. P A N IGRAHI § † Saha Institute of Nucle ar Physics, Bidha nnagar, Kolkata 700 064, India ‡ Dep artment of Physics, T exas A M University, TX 77843, U SA § Physic al R ese ar ch L ab or atory, N avr angpur a, Ahme dab ad 380 009, India E-mail: pr asanta@prl.r es.in URL: http://w ww.prl.r es.in/ ~ prasanta / Received Nov em ber 01, 2006 , in f ina l f orm Marc h 29 , 2007; Published online April 20, 20 07 Original article is a v aila ble at h ttp:/ /www. emis. de/journals/SIGMA/2007/057/ Abstract. A genera l metho d for so lving line a r dif ferential equations o f arbitrar y o rder, is used to a rrive at new representations for the solutions of the known dif ferential equa- tions, b oth without and with a sour ce term. A new quasi- solv able potential has also b een constructed taking recour s e to t he abov e metho d. Key wor ds: Eule r op e r ator; monomia ls ; quas i-exactly s olv a ble mo dels 2000 Mathematics Subje ct Classific ation: 33C99 ; 81 U15 1 A new pro cedure for solving linear dif feren tial equations Linear dif feren tial equations pla y a cru cial role in v arious branc hes of science an d mathematics. Second order dif f er ential equations routinely manifest in the s tudy of quantum mec hanics, in connection with Schr¨ odinger equation. There are v arious tec hniques av ailable to s olve a giv en dif ferentia l equati on, e.g., p o we r serie s metho d , Laplace transform s, etc. Not m an y general metho ds app licable to dif ferentia l equations of arbitrary order exist in the literature (see [1] and references therein). W e make us e of a general metho d for solving linear dif feren tial equations of arbitrary ord er to construct new repr esen tations for the solutions of the kno wn second order linear dif fer ential equations, b oth without and with a source term. This metho d h as found applications in solving linear single and multi v ariable dif f eren tial equations. The so lutions of linear dif ferential equation with a source and devel opmen t of a new Qu asi-Exactly Soluble (QES) system are the new results of th is pap er. 1.1 Case (i). Linear dif ferential equation without a source term After ap p ropriate manipulation, any single v aria ble linear dif ferent ial equation can b e brough t to the follo wing form [ F ( D ) + P ( x, d/dx )] y ( x ) = 0 , (1) where D ≡ x d dx , F ( D ) = P n a n D n is a d iagonal op erato r in the s pace of monomials spann ed b y x n and a n ’s are s ome parameters. Here P ( x, d/dx ) = P i,j c i,j x i ( d dx ) j , where c i,j = 0 if i = j . ⋆ This pap er is a con tribution to the V adim Kuznetsov Memorial Issue ‘Integrable Systems and Related T opics’. The full collection is av ailable at http://www.e mis.de/journals/SIGMA/kuznetso v.html 2 N. Gurapp a, P .K. Jha and P .K. Pa nigrahi Notice that, since F ( D ) is a diagonal op e rator, 1 /F ( D ) is also w ell def ined in t he space of monomials. Th e follo wing ansatz y ( x ) = C λ ( ∞ X m =0 ( − 1) m  1 F ( D ) P ( x, d/dx )  m ) x λ ≡ C λ ˆ G λ x λ (2) is a solution of the ab o v e equation, pr o vided F ( D ) x λ = 0 and th e coef f icien t of x λ in P ( x, d dx ) m x λ should b e zero [6]. In order to r ealise (2) as a p o wer series w e imp ose the r equ iremen t that P ( x, d dx ) lo w ers (i.e. c i,j = 0 for i ≥ j ) or raises the d egree of m onomials. Substituting equa- tion (2), mo dulo C λ , in equation (1) ( F ( D ) + P ( x, d/dx )) ( ∞ X m =0 ( − 1) m  1 F ( D ) P ( x, d/dx )  m ) x λ = F ( D )  1 + 1 F ( D ) P ( x, d/dx )  ( ∞ X m =0 ( − 1) m  1 F ( D ) P ( x, d/dx )  m ) x λ = F ( D ) ∞ X m =0 ( − 1) m  1 F ( D ) P ( x, d/dx )  m x λ + F ( D ) ∞ X m =0 ( − 1) m  1 F ( D ) P ( x, d/dx )  m +1 x λ = F ( D ) x λ − F ( D ) ∞ X m =0 ( − 1) m  1 F ( D ) P ( x, d/dx )  m +1 x λ + F ( D ) ∞ X m =0 ( − 1) m  1 F ( D ) P ( x, d/dx )  m +1 x λ = 0 . Equation (2) co nnects the solution of a giv en dif f eren tial equation to the monomials. In order to show that, this rather straightfo rwa rd pro cedure indeed yields non -trivial results, we explicitly work out a few examples. C onsider the Hermite dif ferenti al equation, whic h arises in the con text of quantum harmonic oscillator,  D − n − 1 2 d 2 dx 2  H n ( x ) = 0 . Here F ( D ) = D − n and F ( D ) x λ = 0 yields λ = n . Hence H n ( x ) = C n ∞ X m =0 ( − 1) m  1 D − n ( − 1 / 2)( d 2 /dx 2 )  m x n . Using [ D , ( d 2 /dx 2 )] = − 2( d 2 /dx 2 ) it is easy to see that  1 ( D − n ) ( − 1 / 2)( d 2 /dx 2 )  m x n = ( − 1 / 2) m ( d 2 /dx 2 ) m m Y l =1 1 ( − 2 l ) x n and H n ( x ) = C n ∞ X m =0 ( − 1 / 4) m 1 m ! ( d 2 /dx 2 ) m x n = C n e − 1 4 d 2 dx 2 x n , this is a w ell-kno wn resu lt. S imilar exp r ession also holds for the Lagurre p olynomial whic h matc hes with the one found in [5]. Belo w, w e list new repr esentati ons for the solutions of some On the Applications of a New T ec hn ique to S olv e Linear DE, w ith and without Sour ce 3 frequent ly encounte red dif feren tial equations in v arious b r anc hes of ph ysics and mathematics [4]. Notice that, the cases of conf lu en t hyp ergeometric, hyp ergeometric, Chebyshev t yp e I I and Jacobi solutions are giv en in [6], and repr o duced here for the sak e of completeness. L e gendr e p olynomial P n ( x ) = C n e −{ 1 / (2[ D + n +1]) } ( d 2 /dx 2 ) x n . Asso ciate d L e gendr e p olynomial P m n ( x ) = C n (1 − x 2 ) m/ 2 e −{ 1 / (2[ D + n + m +1]) } ( d 2 /dx 2 ) x n − m . Bessel function J ± ν ( x ) = C ± ν e −{ 1 / (2[ D ± ν ]) } x 2 x ± ν . Gener alize d Bessel fu nction u ± ( x ) = C ± e −{ β γ 2 / (2[ D + α ± β ν ]) } x 2 β x β ν − α ± β ν . Ge genb auer p olynomial C λ n ( x ) = C n e −{ 1 / (2[ D + n +2 λ ]) } ( d 2 /dx 2 ) x n . Hyp er ge ometric fu nction y ± ( α, β ; γ ; x ) = C ± e −{ (1 / ( D + λ ± ) } ˆ A x − λ ∓ , where λ ± is either α or β and ˆ A ≡ x d 2 dx 2 + γ d dx . All the ab o v e series solutions ha v e descending p o w ers of x . In order t o get th e series in the asc ending p ow ers, one has to replace x by 1 x in the original dif ferential equ ation and g enerate the solutions via equatio n (2). How ever, the n um b er of solutions will r emain the same . One can al so generate the series solutions by multiplying the original dif feren tial equations with x 2 , an d then, rewriting x 2 d 2 dx 2 = D ( D − 1) = F ( D ). The solution for th e follo wing equation with p eriodic p ot en tial d 2 y dx 2 + a cos ( x ) y = 0 (3) can b e f ou n d after multiplying equation (3) b y x 2 and rewriting x 2 d 2 dx 2 as ( D − 1) D to b e y ( x ) = ∞ X m, { n i } =0 ( − a ) m m ! ( m Y i =1 ( − 1) n i (2 n i )! ) ×          m Y r =1  2  m + λ/ 2 − r + m +1 − r P i =1 n i  !  2  m + λ/ 2 + 1 − r + m +1 − r P i =1 n i  !          x 2  m + m P i =1 n i + λ/ 2  , where λ = 0 or 1. In th e same manner, one can w rite do wn the s olutions for the Mathieu’s equation as well. Chebyshev p olynomials T n ( x ) and U n ( x ) T n ( x ) = C n e − n 1 2 1 ( D + n ) d 2 dx 2 o x n 4 N. Gurapp a, P .K. J ha and P .K. P anigrahi and U n ( x ) = C ′ n e − n 1 2 1 ( D + n +2) d 2 dx 2 o x n , where C n and C ′ n are appropr iate normalization constants. Jac obi p ol ynomial J ( α,β ) n ( x ) = ∞ X m =0  1 ( D − n )( D + α + β + 1)  d 2 dx 2 + ( β − α ) d dx  m x n , Sc hl¨ af li, Whittak er and for that matter, an y solution of a second ord er linear dif ferentia l equation without a source term can b e solv ed in a manner id entical to the ab ov e cases. 1.2 Case (ii). Linear dif ferential equation with a source term Consider, ( F ( D ) + P ( x, d/dx )) y ( x ) = Q ( x ) . No w the solution can b e found in a str aigh tforw ard w a y as F ( D )  1 + 1 F ( D ) P ( x, d/dx )  y ( x ) = Q ( x ) , and y ( x ) = ∞ X m =0 ( − 1) m  1 F ( D ) P ( x, d/dx )  m 1 F ( D ) Q ( x ) . W e list b elo w a few examples for the ab o v e case. Neumann ’s p olynomial O n ( x ) = ( ∞ X r =0 ( − 1) r  1 [( D + 1) 2 − n 2 ] x 2  r  1 [( D + 1) 2 − n 2 ]  ) ×  x co s 2 ( nπ / 2) + n sin 2 ( nπ / 2)  . L ommel func tion w ( x ) = ∞ X m =0 ( − 1) m  1 ( D 2 − ν 2 ) x 2  m 1 ( D 2 − ν 2 ) x µ +1 . The cases of S truve , Anger and W eb er functions are id en tical to the ab o v e ones. It is a p riori not transparen t that all the solutions of a giv en dif feren tial equ ation can b e obtained through the present app roac h. If F is a p olynomial of the same degree as the order of the dif f eren tial equ ation, and h as distinct ro ots, the linearly indep endent solutions can b e obtained th rough this approac h. As h as b een p oin ted out in an earlier pap er [8] a give n single v ariable dif feren tial equation can b e cast in the desired form , as requ ired b y the present approac h , in more than one wa y through multiplicati on by p ow ers of x . I n a num b er of cases, these lead to dif ferent solutions. How ever, the case of degenerate ro ots, as also the case of in homogeneous equations needs separate consideration, whic h we hop e to in v estigate in fu tu re. On the Applications of a New T ec hnique to S olv e Linear DE, w ith and without Sour ce 5 2 A new quasi-exactly solv able mo del The ab o v e pr o cedure is also app licable to dif feren tial equations having higher num b er of sin- gularities e.g., Heun equation and its generalizatio n [7]. The same can b e used to generate quasi-solv able mo d els. It is wo rth mentio ning a quasi-solv ab le mo del based on Heun’s equation has b een s tudied; it has b een s ho wn th at, this system lac ks the S L (2 , R ) of many well- kno wn QES mo d el [3, 9]. W e no w p ro ceed to generalize this system for obtaining a new QES p oten tial . The dif fer ential equation under consid eration is give n b y: f ′′ +  1 2 x + 1 + 2 s x − 1 + 1 2( x + ǫ 2 )  f ′ + ( αβ x − q − Ω x ) x ( x − 1)( x + ǫ 2 ) f = 0 . (4) When Ω = 0 the ab ov e reduces to Heun equation. Under the follo wing c hange of v ariable x = sinh 2 ρy 2 1 + 1 ǫ 2 + sinh 2 ρy 2 , and with Ψ = (1 − x ) s f ( x ) the ab ov e equation can b e cast in the form of Sc hr¨ odin ger eigen v alue problem. Here s = (1 − E /ρ 2 ) 1 / 2 , where E is en ergy of the system. Th e constan ts α , β and q in the equation (4 ) are related to s in the follo wing form: α = − 5 2 − s, β = 3 2 − s and q = (1 − s 2 )(1 + ǫ 2 ) − 1 2 sǫ 2 − 1 4 (1 − 2 ǫ 2 ) . The corresp ond ing p oten tial is give n by V ( y ) = ρ 2 ( 8 sin h 4 ρy 2 − 4( 5 ǫ 2 − 1) sinh 2 ρy 2 + 2( 1 ǫ 4 − 1 ǫ 2 − 2) 8(1 + 1 ǫ 2 + sinh 2 ρy 2 ) 2 + Ω ǫ 2 sinh 2 ρy 2 ) . F or the purp ose of f indin g solutions w e cast the generalized Heun equation (4) in the form, [ F ( D ) + P ] y ( x ) = 0. Multiplyin g by x 2 , we get  − 4 x 2 ǫ 2 d 2 dx 2 − 2 ǫ 2 x d dx − 4Ω  f ( x ) +  4 x 3 ( ǫ 2 − 1) d 2 dx 2 + 2(3 ǫ 2 − 2 + 4 sǫ 2 ) x 2 d dx − 4 q x + 4 x 4 d 2 dx 2 + 8(1 + s ) x 3 d dx + 4 αβ x 2  f ( x ) = 0 . In the ab ov e equation F ( D ) = − 4 ǫ 2 D 2 + 2 ǫ 2 D − 4 Ω and the condition F ( D ) x ξ = 0 yields ξ ± = 1 4  1 ± p 1 − 16 Ω /ǫ 2  . F or p olynomial solutions either of the r o ots of the equation must b e an inte ger. T aking ξ − = m w e obtain the follo wing relation for the allo w ed v alues of m 2 m 2 − m + 2Ω /ǫ 2 = 0 . Let us consider a case in wh ic h Ω = − ǫ 2 / 2. T aking care of normalization condition an d p ositivit y of energy , the allo w ed v alue of m is 1. This yields a solution of the form: f ( x ) = a 1 x + a 2 x 2 + a 3 x 3 + · · · . If the p olynomial terminates at x n − 1 then the coef f icien ts a n and a n +1 should b e zero. Using this constrain t, we get the f ollo wing condition for n s 2 + (2 n − 1) + n 2 − n − 15 4 = 0 , 6 N. Gurapp a, P .K. Jha an d P .K. Panigrahi giving the allo w ed v alues of n as 2, 3. F or n = 2, s = 1 / 2, E = 3 4 ρ 2 and ǫ 2 = 1 w e get the wa ve function ψ ( y ) = N  2 2 + sinh 2 ( ρy / 2)  1 / 2  sinh 2 ( ρy / 2) 2 + sinh 2 ( ρy / 2)  , where N is the normalization constan t. F or n = 3, s = − 1 2 , wh ic h makes the wa ve function un-normalizable. This p ro cedure ma y f ind applicabilit y to unrav el the symmetry pr op erties of Heun equation and its generalizat ion [2]. W e hav e earlier stud ied the symmetry p rop erties of conf luent Hyp ergeometric and Hyp ergeometric equations using the pr esen t app roac h [8]. Th e fact that, the solutions are connected with m onomials m akes the searc h for the symmetry r ather straigh tforw ard. In th e case when Ω 6 = 0, in equation (4 ), the additional term in Heun equation manifests in the p oten tial in the Sc hr¨ odinger equation, as a term havi ng 1 x 2 t yp e singularity at the origin. This is very int eresting since it is of the Calogero–Sutherland t yp e, a system muc h inv estigated in the literature. Like the Calogero–Sutherland case, a Jastro w t yp e factor in the w a v e fu nction app ears b ecause of th is singular in teraction. At a formal level, this int eraction mo dif ies the singularit y at th e origin without adding any new singularit y . W e note that x = 0 is still a regular singular p oin t, s ince the limit x 2 Ω x 2 as x → 0 is f inite. In conclusion, w e h a v e dev elop ed a n ew m etho d to solv e linear dif ferential equations of arbitrary order, b oth without and with source term. Using this metho d , we w ork ed out a few examples for the case of second order linear d if ferentia l equations. W e obtained kno wn, as w ell as, the new representa tions for the corresp onding solutions. The same approac h is also applicable to Q ES mo d els based on Heun equation and its generalizations. In particular a new p oten tial of Q ES type is constructed th rough this app roac h. W e in tend to analyze the s ymmetry prop erties of the Heun equation through th e p resen t formalism. Ac kno wledgemen ts W e thank C. Sud heesh for giving a careful reading to the manuscript and Vive k Vy as for many useful comment s. References [1] Ad omian G., S olving F rontie r Problems of physics: the d ecomp osition metho d, Kluw er, Dordrech t, 1994. [2] Atre R., Mohapatra C.S., Panig rahi P .K., Finding exact and app ro ximate wa ve functions of Ho oke’s atom, their information entrop y and correlation, Phys. L ett. A 361 (2007), 33–38. [3] Christ N.H., Lee T.D., Quantum expansion of soliton solutions, Phys. R ev. D 12 (1975), 1606–1627. [4] Gradshteyn I .S., Ryzhik I .M., T ables of integra ls, series and pro ducts, Academic Press I nc., 1965. [5] Gurapp a N ., Khare A., P anigrahi P .K., Connection betw een Calogero –Marc hioro–W olfes typ e few-b o dy mod els and free oscillators, Phys. L ett . A 244 (1998), 467–472, cond -mat/9804207. 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