New Quasi-Exactly Solvable Difference Equation

Abstract

Exact solvability of two typical examples of the discrete quantum mechanics, i.e. the dynamics of the Meixner-Pollaczek and the continuous Hahn polynomials with full parameters, is newly demonstrated both at the Schrödinger and Heisenberg picture levels. A new quasiexactly solvable difference equation is constructed by crossing these two dynamics, that is, the quadratic potential function of the continuous Hahn polynomials is multiplied by the constant phase factor of the Meixner-Pollaczek type. Its ordinary quantum mechanical counterpart, if exists, does not seem to be known.

Exactly and quasi-exactly solvable ‘discrete’ quantum mechanics

“…Many examples are known in oQM [40][41][42][43] but only a few are known in dQM [44,45] in spite of the proposal that the sl(2, R) algebra characterization of quasi-exact solvability could be extended to difference Schrödinger equation [46]. This unified theory also incorporates the known examples of QES Hamiltonians in dQM [47][48][49]. A new type of QES Hamiltonians is constructed.…”

“…Known discrete QES examples belong to this class [48,49]. For the L = 4 case, H is defined by adding a linear and a quadratic in h(x) compensation terms to the Hamiltonian H: H def = H − e 0 (M )h(x) 2 − e 1 (M )h(x), (3.10) and H h(x) M ∈ V M .…”

Bethe Ansatz Solutions to Quasi Exactly Solvable Difference Equations

“…This is in sharp contrast to the sl(2, R) characterisation [13], whose applicability is limited to essentially single degree of freedom systems. Recently the deformation procedure was applied to exactly solvable 'discrete' quantum systems of one and many degrees of freedom to obtain corresponding 'discrete' QES systems [10,11,16]. In this paper we present Bethe ansatz solutions for these QES difference equations.…”

“…In this paper we present Bethe ansatz formulations and solutions for a family of Quasi-Exactly Solvable (QES) difference equations, which were recently introduced by one of the present authors [10,11]. One of the main purposes of the present paper is to provide a good list of explicit Bethe-ansatz equations for the quasi-exactly solvable difference equations (2.11), (3.11), (3.15), (4.9), (4.17), and (5.5).…”

“…To our knowledge, the Bethe ansatz solutions were known only for some QES difference equations in connection with U q (sl(2)) [17], and for exactly solvable difference equations [18,19,20] of the elliptic Rusijsenaars-Schneider system. Here we apply the Bethe ansatz formulation to several explicit examples of QES difference equations [10,11] as deformation of exactly solvable 'discrete' quantum mechanics [2,3], which are difference analogues of the well-known quasi exactly solvable systems, the harmonic oscillator (with/without the centrifugal potential) deformed by a sextic potential and the 1/ sin 2 x potential deformed by a cos 2x potential. As will be shown explicitly in the main text, these Bethe ansatz equations (2.11), (3.11), (3.15), (4.9), (4.17), and (5.5) can be considered as deformations of the equations (2.16), (4.14), (5.9) determining the roots of the corresponding (q-)Askey scheme of hypergeometric orthogonal polynomials [9] (the MeixnerPollaczek, continuous Hahn, continuous dual Hahn, Wilson and Askey-Wilson polynomials and their restrictions) which constitute the eigenfunctions of the undeformed exactly solvable quantum systems.…”