Charged particles in external fields as physical examples of quasi-exactly-solvable models: A unified treatment

Abstract

We present a unified treatment of three cases of quasi-exactly-solvable problems, namely, a charged particle moving in Coulomb and magnetic fields for both the Schrödinger and the Klein-Gordon case, and the relative motion of two charged particles in an external oscillator potential. We show that all these cases are reducible to the same basic equation, which is quasiexactly solvable owing to the existence of a hidden sl 2 algebraic structure. A systematic and unified algebraic solution to the basic equation using the method of factorization is given. Analytical expressions of the energies and the allowed frequencies for the three cases are given in terms of roots of one and the same set of Bethe ansatz equations.

A unified treatment of polynomial solutions and constraint polynomials of the Rabi models

“…The point of crucial importance is that with E being fixed by (38), all the coefficients of the linear 2nd order ODE (37) are fixed, and thus there are only two linearly independent solutions of Eq. (37), with at most one of them being polynomial one.…”

“…The remaining sl 2 algebraization conditions are satisfied by C 00 = a 2 = 1, C 0 = −2ǫ+2j −1, together with the relation (27) In the case of a relative motion of two electrons in an external oscillator potential [36,37] one had an ordinary eigenvalue problem Lψ = λψ (cf. Eq.…”

Exact Solutions of a Class of Double-Well Potentials: Algebraic Bethe Ansatz

“…In this section, we introduce the three DWPs that are discussed in this work and solve the corresponding Schrödinger equations via the factorization method in the framework of algebraic Bethe ansatz [15]. The general exact expressions for the energies, the wave functions, and the allowed values of the potential parameters are obtained in terms of the roots of the Bethe ansatz equations.…”

“…The fundamental idea behind the quasi-exact solvability is the existence of a hidden dynamical symmetry. QES systems can be studied by two main approaches: the analytical approach based on the Bethe ansatz [14][15][16][17][18][19] and the Lie algebraic approach [10][11][12][13]. These techniques are of great importance because only a few number of problems in quantum mechanics can be solved exactly.…”

Spinless Particles in Exponentially Varying Electric and Magnetic Fields

“…Fundamental processes occurring in these applications, developments on the intense particle beams generated by sources such as lasers, nuclear reactors, or accelerators, and appearance of the different configurations of parallel electric and magnetic fields in astrophysics dating from the discovery of the pulsars required noticeable interest on the exact solutions of the relativistic particle equations in external electromagnetic fields. Such efforts have been performed for different configurations of the external fields [1][2][3][4][5][6][7][8][9][10]. In these studies, the exact solutions of the nonrelativistic and relativistic wave equations have provided important information regarding the quantum mechanical system.…”