Unified theory of exactly and quasi-exactly solvable `Discrete quantum mechanics: I. Formalism

The second is the closure relation [5]. It allows to construct the exact Heisenberg operator solution of the sinusoidal coordinate η(x), which generates the closure relation together with the Hamiltonian. The positive/negative energy parts of the Heisenberg operator solution give the annihilation/creation operators, in terms of which every eigenstate can be built up algebraically starting from the groundstate. Thus exact solvability in the Heisenberg picture is realised.

It is interesting to note that these two sufficient conditions apply equally well in the ‘discrete’ quantum mechanics (QM) [6,7,5,8,9], which is a simple extension or deformation of QM. In discrete QM the dynamical variables are, as in the ordinary QM, the coordinate x and the conjugate momentum p, which is realised as p = -i∂ x . The Hamiltonian contains the momentum operator in exponentiated forms e ±βp , which acts on wavefunctions as finite shift operators, either in the pure imaginary directions or the real directions. Thus the Schrödinger equation in discrete QM is a difference equation instead of differential in ordinary QM. Various examples of exactly solvable discrete quantum mechanics are known for both of the two types of shifts [6,7,10,5,8,9], and the eigenfunctions consist of the Askey-scheme of the hypergeometric orthogonal polynomials [11,12,13] of a continuous (pure imaginary shifts) and a discrete (real shifts) variable.

It should be stressed, however, that these two sufficient conditions do not tell how to build exactly solvable models. In this paper we present a simple theory of constructing exactly solvable Hamiltonians in discrete QM. It covers all the known examples of exactly solvable discrete QM with both pure imaginary and real shifts [8,9] and it predicts several new ones to be explored in a subsequent publication [14]. Moreover, the theory is general enough to generate quasi-exactly solvable Hamiltonians in the same manner. The quasiexact solvability means, in contrast to the exact solvability, that only a finite number of energy eigenvalues and the corresponding eigenfunctions can be obtained exactly [15]. This unified theory also incorporates the known examples of quasi-exactly solvable Hamiltonians [16,17]. A new type of quasi-exactly solvable Hamiltonians is constructed in this paper and its explicit examples will be surveyed in a subsequent publication [14]. One of the merits of the present approach is that it reveals the common structure underlying the exactly and quasi-exactly solvable theories. In ordinary QM, the corresponding theory was already given in the Appendix A of [5], although it does not cover the quasi-exact solvability.

The present paper is organised as follows. In section two the general setting of the discrete quantum mechanics is briefly reviewed and in §2.1 the Hamiltonians for the pure imaginary shifts and for the real shifts cases are given and the general strategy of working in the vector space of polynomials in the sinusoidal coordinate is explained. In §2.2, based on a few postulates, various properties of the sinusoidal coordinate η(x), which is the essential ingredient of the present theory, are presented in some detail. The main result of the paper, the unified form of the exactly and quasi-exactly solvable ‘Hamiltonians, ’ is given in §2. 3. The action of the Hamiltonian on the polynomials of the sinusoidal coordinate is explained in §2. 4. It simply maps a degree n polynomial into a degree n + L -2 polynomial. Here L is the degree of a certain polynomial constituting the potential function in the Hamiltonian. The exactly solvable case (L = 2) is discussed in section three. In §3.1, the closure relation, which used to be verified for each given Hamiltonian, is shown to be satisfied once and for all by the proposed exactly solvable Hamiltonian. The nature of the dual closure relation, which plays an important role in the theory of discrete QM with real shifts and the corresponding theory of orthogonal polynomials of a discrete variable, is examined and compared with that of the closure relation in §3.2. The relationship between the closure plus dual closure relations and the Askey-Wilson algebra [18,19,20,21] is elucidated in §3.3. In §3.4, shape invariance is explained and shown to be satisfied for the pure imaginary shifts case §3.4.1 and for the real shifts case §3.4.2. The quasi-exactly solvable ‘Hamiltonians’ are discussed in section four.

The QES case with L = 3 is achieved in §4.1 by adjusting the compensation term which is linear in

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