The q-deformation of Hyperbolic and Trigonometric Potentials

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1 International Journal of Difference Euations ISSN , Volume 9, Number 1, pp The -deformation of Hyperbolic and Trigonometric Potentials Alina Dobrogowska Institute of Mathematics, University of Białystok Akademicka 2, Białystok, Poland Abstract We present solutions of -deformed Schrödinger euation for the hyperbolic and trigonometric potentials given by a factorization method. Their various properties including the correspondence 1 to the non deformed Schrödinger operators are discussed. AMS Subject Classifications: 39A70, 39A13. Keywords: -difference operators, factorization, -hyperbolic and -trigonometric potentials. 1 Introduction The paper deals with a second order -difference euation such that in the limit 1 we get a differential euation related to the stationary Schrödinger euation with the Scarf potentials [14]. We shall consider eigenproblems H k ψ k = λ k ψ k, 1.1 for the discrete family k N {0} of the second order -difference operators H k = Z k x Q 1 + W k x + V k x, k N {0} 1.2 acting in the Hilbert spaces H k, where 0 < < 1 and is the -derivative see [9] ψx = ψx ψx x Received November 30, 2013; Accepted March 15, 2014 Communicated by Malgorzata Migda

2 46 Alina Dobrogowska By definition, H k consists of the complex valued functions ψ : [a, b] C defined on the -interval [a, b] := { n a : n N {0}} { n b : n N {0}}, a, b R, a < b, a n b, which are suare integrable with respect to the scalar products ψ ϕ k := b Let us recall that the -integral is given by b a ψxd x := ψxd x = n= a ψxϕxϱ k xd x n bψ n b aψ n a, 1.5 n=0 1 n ψ n + and the shift operators Q, Q 1 are given by n= 1 n ψ n Qψx = ψx, 1.6 Q 1 ψx = ψ 1 x. 1.7 We postulate some conditions which must be satisfied by weight functions, ϱ k 1 = Q B k ϱ k, 1.8 B k ϱ k = A k ϱ k, 1.9 where A k, B k are real valued functions on [a, b]. The euation 1.9 corresponds in the limit 1 to the Pearson euation, which is important for the theory of classical orthogonal polynomials. Additionally we impose the boundary conditions B k aϱ k a = B k bϱ k b = The basic idea of the factorization method is well known, see [10 12]. Some results concerning the factorization method for the second order -difference euations were presented for example in papers [3, 13]. In this work, we apply our previous results obtained in the paper [6]. We will factorize the chain of second order -difference operators 1.2 using the first order -difference operators of the form A k = + f k, 1.11 A k = + f k = B k Q 1 + f k Ak xf k The operator A k : H k 1 H k is adjoint to A k : H k H k 1 with respect to the scalar products 1.4. We say that the operators H k admit a factorization if there exist seuences of operators A k, A k and constants a k such that H k = A ka k + a k = A k+1 A k+1 + a k+1 for k N {0}. 1.13

3 The -deformation of hyperbolic and -trigonometric potentials 47 In papers [5, 7] we considered the -deformation of the Schrödinger operator with the following potentials: shifted oscillator, isotropic oscillator, Rosen Morse II, Eckart II, Poschl Teller I and II. Next in the paper [8] we considered the case of -deformed Schrödinger euation for the Morse potential. In this work we will study the case of -deformed Schrödinger euation for the -hyperbolic and -trigonometric Scarf potentials. 2 The -hyperbolic and -trigonometric Potentials We obtain the case of the -deformation of hyperbolic and trigonometric potentials if we consider the operators H k in the following forms: H k = k b 2 + a 0 a 1 b 2 x 2 1 2k 1 x h k 1 x + b + b b 2 2k 2 x 2 Q 1 + b 0 k + 1 x b 2 + a 0 a 1 1 2k+1 x h k x + b 1 0 b 2 2k x 2 + b 0 b 2 + a 0 a 1 b 2 x 2 1 2k 1 x h k 1 x + b + b b 2 2k 2 x 2 + b k+1 b 2 x 2 + b x 2 b 2 + a 0 a 1 1 2k 1 x h k 1 x + b 1 0 b 2 2k 2 x 2 + b 0 k 1 2 x 2 b 2 + a 0 a 1 1 2k+1 x h k x + b 1 0 b 2 2k x 2 + b 0 k x 2 b 2 + a 0 a 1 2k+1 x h k x + b k+1 b 2 x 2 + b x k a 1 [k] a 0 [k 1] b 2 [k 1] [k],

4 48 Alina Dobrogowska where [k] = 1 k 1. This chain is factorized using first order -difference operators 1 A k = 1 x Q + A k = k b 2 x 2 + b 0 1 x Q 1 b 2 + a 0 a b 2 + a 0 a 1 2k+1 x h k x + b 1 1 0, x 2 b 2 2k x 2 + b 0 b2 2k x 2 + b k 1 x 2k+1 x h k x + b 0. The constants a k in the factorization relations 1.13 are given by the following formula a k = 1 k a 1 [k] a 0 [k 1] b 2 [k 1] [k]. 2.4 In this case the functions B k, A k and f k are given by B k x = k b 2 x 2 + b 0, 2.5 A k x = k [ 2k] b 2 x, 2.6 b 2 + a 0 a 1 2k+1 x h k x + b 1 f k x = x 2 b 2 2k x 2 + b 0 1 x. 2.7 Solving the euation 1.9 we obtain the weight function ϱ k x = 1, 2.8 2k x 2 b 2 b0 ; k+1 2 where x; 2 k+1 = 1 x1 2 x k x. Substituting 2.5 and 2.8 into the boundary condition 1.10 we choose as the - interval [a, b] = [, ]. It is easy to see that any solution of the euation A k ψ 0 k = is automatically a solution of euation 1.1 with the eigenvalue λ k = a k. Solving the euation 2.9 we obtain ψkx 0 b = C 2 2i x 2 + b 0, 2.10 i= k b 2 + a 0 a 1 1 2i+1 x h i x + b 1 0

5 The -deformation of hyperbolic and -trigonometric potentials 49 where C R. The solutions 2.10 can be used to construct solutions ψ k+n = A k+n... A k+1ψ 0 k of the eigenvalue problems for the operators H k+n given by Limit Case The second order -difference operator 2.1 in the limit case as 1 becomes the second order differential operator H k = b 2 x 2 + b 0 d2 dx 2 + 2kb 2x d dx + 2kb2 2x 2 + kb 2 a 0 a 1 x 2 + kb 2 hx b 2 x 2 + b 0 + a 0 a 1 x 2 + 2ha 0 a 1 x + h 2 + 4b 2 b 0 4b 2 x 2 + b 0 + ka 1 k 1a 0 kk 1b a 0 a The chains of operators 1.11 and 1.12 in the limit are given by A k = d dx + 2b 2x + a 0 a 1 x + h, b 2 x 2 + b 0 A k = b 2 x 2 + b 0 d dx + 2k 1b 2x + a 0 a 1 x + h The constants a k 2.4 are transformed into a k = ka 1 k 1a 0 kk 1b The ground state 2.10 tends to ψkx 0 = b 2 x 2 + b 0 2b 2 +a 1 a 0 4b 2 and the weight function 2.8 to exp h 2b 0 dx 1 + b 2 b0 x 2 ϱ k x = b 2 x 2 + b 0 k+1. In order to systematize the class of the potentials given by -deformation we shall transform the considered differential operator 3.1 into the standard form where H k = d2 dy 2 + V kxy, 3.5 V k x = d 2x + d 1 b 2 x 2 + b 0 + d 0 3.6

6 50 Alina Dobrogowska and d 2 = h kb 2 + a 0 a 1, d 1 = h b 2b 0 k b 2 + a 0 a 1 k + 1 b 0, b 2 d 0 = a 0 a 1 + a 1 + a b b The transformation connecting these operators and these eigenvalue problems are following ψ k x = b 2 x 2 2k b 0 ϕ k y, 3.10 dx dy =, b2 x 2 + b where ϕ k is eigenfunction of the operator 3.5. These functions belong to the standard Hilbert space L 2 R, dx. For the different values of the parameters b 2, b 0 we have the following possibilities: 1. If b 2, b 0 > 0 from 3.11 we obtain b0 x = sinh b2 y c, b 2 where c is constant. In this subcase the potential is given by V k y = d 1 cosh 2 y c + d 2 sinhy c cosh 2 y c + d 0, where we put b 2 = b 0 = 1. We see that in the limit we obtain the hyperbolic potential [1,2,4]. This potential was proposed and solved by F. Scarf. It is usually called the hyperbolic Scarf potential. The domain for this potential is,. 2. If b 2 < 0 and b 0 > 0 from 3.11 we obtain b0 x = sin b 2 y c. In this subcase the potential is given by V k y = b 2 d 1 cos 2 y c + d 2 siny c cos 2 y c + d 0, where we put b 2 = b 0 = 1. We see that in the limit we obtain the trigonometric potential [1,2,4]. In the literature it is know as trigonometric Scarf potential. The domain for this potential is π 2 + c, π 2 + c.

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