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One-parameter Isospectral Special Functions M.A.Reyes , D.Jimenez and H.C.Rosu † † ‡ Instituto de F´ısica, Universidad de Guanajuato, Apdo. Postal E143, 37150 Leoń,Gto., México † Applied Mathematics, IPICyT, Apdo. Postal 3-74 Tangamanga, San Luis Potos´ı, México ‡ 3 0 0 Abstract. Using a combination of the ladder operators of Pinã [Rev. Mex. F´ıs. 2 41 (1995) 913] and the parametric operators of Mielnik [J. Math. Phys. 25 (1984) n 3387] we introduce second order linear differential equations whose eigenfunctions a J are isospectral to the special functions of the mathematical physics and illustrate 5 the method with several key examples. 1 Resumen. Usando una combinacioń de los operadores de escalera de Pinã [Rev. 1 v Mex. F´ıs. 41 (1995) 913] y de los operadores parametricos de Mielnik [J. Math. 2 Phys. 25 (1984) 3387] introducimos operadores lineales de segundo orden con 2 0 eigenfunciones que son formas isoespectrales de las funciones especiales de la f´ısica 1 matema´tica y presentamos algunos ejemplos ba´sicos. 0 3 0 1 Introduction / h p Theuse ofthefactorizationmethod[1]proved tobeapowerful toolforextending theclass - h of exactly solvable Sturm-Liouville problems especially in quantum mechanics, where in t a the form of supersymmetric quantum mechanics led to new potentials, which are isospec- m tral to a given problem [2]. In a paper by Mielnik [3], the usual factorization operators : v i 1 d 1 d X a = +x a = +x (1) ∗ r √2 dx ! √2 −dx ! a of the one dimensional harmonic oscillator Hamiltonian 1 1 d2 1 1 H + = + x2 + (2) 2 −2dx2 2 2 have been replaced by new operators 1 d 1 d b = +β(x) b = +β(x) . (3) ∗ √2 dx ! √2 −dx ! In order that these new operators factorize the same Hamiltonian (2) 1 bb = aa = H + (4) ∗ ∗ 2 the function β(x) should satisfy a Riccati equation of the form β +β2 = 1+x2 . (5) ′ 1 Using the solutions of this equation e x2 − β(x) = x+ (6) x γ + e y2dy − 0 R one can introduce a new Hamiltonian H, which is defined by the inverse factorization of (4) f 1 1 d e x2 b b = H = H + 1  −  (7) ∗ − 2 2 − − dx x γ + e y2dy  −  f  0   R  with new potential functions x2 d e x2 V(x) =  −  (8) 2 − dx x γ + e y2dy  −  e  0   R  and whose eigenfunctions ψ = b ψ (n = 1,2,...) (9) n ∗ n 1 − are isospectral to the harmonice oscillator eigenfunctions ψ . n 2 Factorization of special functions The great majority of differential equations appearing in Physics can be factorized by means of ladder operators. Therefore, we should be able to apply the procedure de- scribed in the previous section to the raising and lowering operators of the important class of Sturm-Liouville problems and get in this way isospectral second order differential equations. To attain this objective we proceed as follows. Oneofthepossible formsoffactorizingasubclass ofsecondorder differential operators associated to the special functions of mathematical physics was introduced by Pinã [4]. Consider the Sturm-Liouville problem d2 d ψ (x) P(x) +Q(x) +R (x) ψ (x) = 0 , (10) Ln n ≡ " dx2 dx n # n where P, Q and R are functions of the variable x, and R depend on the index n. Then, n n it is possible to construct raising and lowering operators [4] d d A+ = √P +a+ A = √P +a (11) n dx n −n dx −n 2 that can factorize Eq.(10) in two ways A A+ = +K (12) −n+1 n Ln n A+A = +K , (13) n −n+1 Ln+1 n where the constant K is the same in both factorizations. Furthermore, the functions a+, n n a , turn out to be −n+1 1 Q d 1 a = √P +c + (R R )dx (14) −n+1 2 "√P − dx n √P n+1 − n # Z 1 Q d 1 a+ = √P c (R R )dx , (15) n 2 "√P − dx − n − √P n+1 − n # Z where c is an integration constant. From Eq.(12), one may consider the constant K as n n the eigenvalue corresponding to the eigenfunction ψ for the operator A A+. n −n+1 n Let us now define new operators B+, B by n n−+1 B+ = A+ +b+ n n n B = A +b (16) n− −n −n and demand that they can also factorize the Sturm-Liouville operator (x) as n L B B+ = A A+ = (x)+K . (17) n−+1 n −n+1 n Ln n Then, the following relationship should be fulfilled d d db+ √P b+ +b +√P n +b+a +a+b +b+b = 0 . (18) n dx −n+1dx dx n −n+1 n −n+1 n −n+1 ! Therefore, the functions b+, b must satisfy n −n+1 b = b+ (19) −n+1 − n db+ √P n b+2 +b+(a a+) = 0 . (20) dx − n n −n+1 − n Eq.(20) is a Riccati type equation, which can be easily solved to get eδ(x) b+(x) = . (21) n x γ eδ(y) dy −x0 √P(y) R Here, δ(x) is defined by the indefinite integral x (a+(y) a (y)) δ(x) n − −n+1 dy (22) ≡ P(y) Z q 3 and γ is an integration constant. x may be chosen as the point where the integrand 0 vanishes. Similarly to Mielnik’s new Hamiltonian H, we now introduce the second order differential operator (x) given by n L f e db+ = B+B K = 2√P n . (23) Ln+1 n n−+1 − n Ln+1 − dx e Thatis, thenewoperator differsfrom bythederivativeofthesolutionoftheRiccati n n L L equation (20), in the same way as Mielnik’s Hamiltonian H differs from the harmonic e oscillator Hamiltonian H, as can be seen in Eq.(7). f If we now define the functions ψ B+ψ n = 0,1,2,... (24) n+1 ≡ n n where ψ are the eigenfunctioens of , we can see that n n L ψ = B+B K B+ψ = B+ ψ = 0 (25) Ln+1 n+1 n n−+1 − n n n nLn n (cid:16) (cid:17) e e Therefore, these ψ are the eigenfunctions for the new operator , and we can write the n n L eigenvalue equation e e B+B ψ = +K ψ = K ψ (26) n n−+1 n+1 Ln+1 n n+1 n n+1 (cid:16) (cid:17) e e e AscanbeseenfromEq.(21),wehaveconstructedaoneparamenterfamilyofoperators (x;γ) which are “isospectral” to the original Sturm-Liouville operator (x), for the n n L L allowed values of the parameter γ that produce a non-divergent function b+(x). n e As we can see, B+, B are not ladder operators as are A+, A . However, one can n n−+1 n −n+1 easily verify that the third order operators C+ B+A+ B (27) n ≡ n n−1 n− C B+ A B (28) n−+1 ≡ n 1 −n n−+1 − play the role of raising and lowering operators, respectively, for the functions ψ (x). The n use of ladder operators of order higher than two is not easy to find in the literature, but e some work in this direction has already been reported [5]. Since B , and therefore C , is not defined forn=0, the space defined by the eigen- n−+1 n−+1 functions ψ lack the element with n=0, similarly to what happens in SUSY factorization n [2]. e 3 Examples Here, we proceed to find the new operators (x;γ) and their eigenfunctions, from the n L factorizations of the special functions of mathematical physics. e 4 3.1 Hermite polynomials The Hermite differential equation d2H (x) dH (x) n 2x n +2nH (x) = 0 (29) dx2 − dx n has raising and lowering differential operators given by d 2x H (x) = H (30) n n+1 dx − ! − d H (x) = 2(n+1)H (x) . (31) dx n+1 n In this case, the δ-integral in Eq.(22) is x δ = ( 2y)dy = x2 − − Z and therefore e x2 b+ = − . (32) n x γ e y2dy − − 0 R The integrand in thedenominator being positive definite, one should impose the condition γ > √π [3] in order to have a well-defined operator. | | 2 Now, with the use of Eq.(20) one gets 2 db+ e x2 2xe x2 n =  −  − , (33) dx x − x γ e y2dy γ e y2dy  −  −  −  −  0  0  R  R and therefore, the second order differential operator d2 d 4xe x2 2e 2x2 − − (x;γ) = 2x +2n+ (34) Ln+1 dx2 − dx γ xe y2dy − x 2 − γ e y2dy e −R0 −0 − ! R has parametric eigenfunctions given by e x2 − H (x;γ) = H (x)+ H (x) . (35) n+1 − n+1 x n γ e y2dy − f −0 R 5 3.2 Laguerre polynomials For the Laguerre differential equation d2Lα(x) dLα(x) x2 n +[(α+1)x x2] n +nxLα(x) = 0 , (36) dx2 − dx n the raising and lowering operators are d x +α+n+1 x Lα(x) = (n+1)Lα (x) (37) dx − ! n n+1 d x n 1 Lα (x) = (α+n+1)Lα(x) . (38) dx − − ! n+1 − n The δ-integral is x α+2(n+1) y δ = − dy = lnx[α+2(n+1)] x (39) y − Z and hence xα+2n+2e x b+ = − . (40) n x γ y(α+2n+1)e ydy − − 0 R For x > 0, the integral in the denominator is positive definite, with maximum value ∞ y(α+2n+1)e ydy = Γ(α+2n+2) − Z 0 and, since there is no upper limit for increasing n, we must have γ < 0. The second order differential operator d2 d 2[x (α+2n+2)]xα+2n+1e x (x;γ) = x2 + (α+1)x x2 +nx+ − − Ln+1 dx2 − dx x γ yα+2n+1e ydy h i − e −0 2x2α+4n+2e 2x R − (41) − x 2 γ yα+2n+1e ydy − −0 ! R has as parametric eigenfunctions xα+2n+2e x Lα (x;γ) = (n+1)Lα (x)+ − Lα . (42) n+1 n+1 x n γ yα+2n+1e ydy − e −0 R 6 3.3 Legendre polynomials The Legendre differential equation d2P (x) dP (x) (x2 1)2 n +2x(x2 1) n n(n+1)(x2 1)P (x) = 0 (43) − dx2 − dx − − n has the raising and lowering operators d (x2 1) +(n+1)x P (x) = (n+1)P (x) (44) n n+1 " − dx # d (x2 1) (n+1)x P (x) = (n+1)P (x) . (45) n+1 n " − dx − # − The δ-integral is in this case x y δ = 2(n+1) dy = ln(1 x2)n+1 (46) − 1 y2 − Z − hence (1 x2)n+1 b+ = − . (47) n x γ (1 y2)ndy − − 1 −R The integral in the denominator is positive definite, which, for a given n, has maximum value +1 π/2 2(2n)!! (1 x2)ndx = 2 sen2n+1θdθ = − (2n+1)!! Z Z 1 0 − and, therefore, γ > 2. | | Hence, the second order differential operator d2 d (x;γ) = (1 x2) 2x 1 x2 +n(n+1)(1 x2)+ Ln+1 − dx2 − − dx − (cid:16) (cid:17) e 4(n+1)(1 x2)n+1 2(1 x2)2n+2 − − (48) γ x(1 y2)ndy − x 2 γ (1 y2)ndy − − −R1 −−R1 − ! has as parametric eigenfunctions (1 x2)n+1 P (x;γ) = (n+1)P (x)+ − P (x) . (49) n+1 − n+1 x n γ (1 y2)ndy e − 1 − −R 7 3.4 Chebyshev polynomials For the Chebyshev differential equation d2T (x) dT (x) (1 x2)2 n x(1 x2) n +n2(1 x2)T (x) = 0 , (50) − dx2 − − dx − n the corresponding raising and lowering operators are d (1 x2) nx T (x) = nT (x) . (51) n n+1 " − dx − # − d (1 x2) +(n+1)x T (x) = (n+1)T (x) . (52) n+1 n " − dx # In this case, the δ-integral is x y 1 δ = (2n+1) dy = n+ ln(1 x2) (53) − 1 y2 2 − Z − (cid:18) (cid:19) and, hence (1 x2)n+1 b+ = − 2 . (54) n x γ (1 y2)n 1dy −2 − − 1 R The integral in the denominator has maximum value 1 π/2 π(2n 1)!! (1 x2)n−21dx = 2 sen2nθdθ = − , − (2n)!! Z Z 1 0 − which imposes the condition γ > π. We can thus construct the second order differential operator d2 d (x;γ) = (1 x2) x 1 x2 +n2(1 x2)+ Ln+1 − dx2 − − dx − (cid:16) (cid:17) e 2(2n+1)x(1 x2)n+21 2(1 x2)2n+1 − − (55) γ x(1 y2)n−21dy − γ x(1 y2)n 1dy 2 − − −2 −R1 −−R1 − ! with the parametric eigenfunctions (1 x2)n+1 2 T (x;γ) = nT (x)+ − T (x) . (56) n+1 − n+1 γ x(1 y2)n 1dy n −2 e − 1 − −R 8 3.5 Jacobi functions The equation defining the Jacobi functions d2f (x) df (x) x2(1 x)2 n +x(1 x)[λ (α+1)x] n +x(1 x)n(n+α)f (x) = 0 (57) − dx − − dx − n has the raising and lowering operators d n+λ (n+α)(n+λ) x(1 x) (n+α) x f (x) = f (x) (58) " − dx − − 2n+α+1!# n 2n+α+1 n+1 d n+1+α λ (n+1)(n+1+α λ) x(1 x) +(n+1) x − f (x) = − f (x) . n+1 n " − dx − 2n+α+1 !# − 2n+α+1 (59) Hence, the δ-integral becomes x u vy δ = − dy = ulnx+(v u)ln(1 x) , y(1 y) − − Z − where (n+α)(n+λ)+(n+1)(n+1+α λ) u = − , v = 2n+α+1 . 2n+α+1 Therefore, we find that xu(1 x)v u b+ = − − , (60) n x γ yu 1(1 y)v u 1dy − − − − − where the integral in the denominatoRr can be written in terms of the incomplete Beta functions B (u,v u) and B (u,v u). In the particular case v > u, to obtain the range 1 x − − − of allowed values for the γ parameter, we see that 1 1 1 − xu 1(1 x)v u 1dx = xu 1(1 x)v u 1dx+ xu 1(1 x)v u 1dx < − − − − − − − − − − − − − Z Z Z 1 0 0 − 1 2Γ(v u)Γ(u) < 2 xu 1(1 x)v u 1dx = 2B(u,v u) = − . (61) − − − − − Γ(v) Z 0 Therefore, we have to demand γ > 2Γ(v−u)Γ(u). | | Γ(v) The second order differential operator d2 d (x;γ) = x2(1 x2) +x(1 x)(λ (α+1)x) +x(1 x)n(n+α)+ Ln+1 − dx2 − − dx − e 2(vx u)xu(1 x)v−u 2x2u(1 x)2(v−u) + − − − (62) γ x yu 1(1 y)v u 1dy − x 2 − − − γ yu 1(1 y)v u 1dy −−R1 − −−R1 − − − − ! 9 has the parametric eigenfunctions (n+α)(n+λ) xu(1 x)v u − f (x;γ) = f (x)+ − f (x) . (63) n+1 2n+α+1 n+1 x n γ yu 1(1 y)v u 1dy − − − e − 1 − −R 3.6 Jacobi polynomials The differential equation defining the Jacobi polynomials d2Pαβ(x) dPαβ(x) (1 x2)2 n + (1 x2)[β α (α+β +2)x] n + − dx2 − − − dx +(1 x2)n(n+α+β +1)Pαβ(x) = 0 (64) − n has raising and lowering operators given by d β α (1 x2) + (n+1+α+β) x+ − Pαβ(x) = " − dx − 2n+2+α+β!# n 2(n+1)(n+1+α+β) = Pαβ (x) (65) − 2n+2+α+β n+1 d β α (1 x2) + (n+1)(x+ − ) Pαβ (x) = " − dx 2n+2+α+β # n+1 2(n+1+α)(n+1+β) = Pαβ(x) . (66) 2n+2+α+β n In this case, we have x p qy 1 1 δ = − dy = (q +p)ln(1+x)+ (q p)ln(1 x) , 1 y2 2 2 − − Z − where β2 α2 p = − , q = 2n+2+α+β . 2n+2+α+β Hence (1+x)1(q+p)(1 x)1(q p) b+ = 2 − 2 − . (67) n x γ (1+y)1(q+p) 1(1 y)1(q p) 1dy 2 − 2 − − − − 1 −R For the parameter γ, since q > p, we demand that 1 Γ(q+p)Γ(q p) γ > (1+x)12(q+p)−1(1 x)21(q−p)−1dx = 2q−1 2 −2 . − Γ(q) Z 1 − 10