Exactly solvable generalized quantum harmonic oscillators related with the classical orthogonal polynomials

Swapan Kumar Mandal

Xuehong Wang, Yubin Liu

Siyu Li, Ian Marquette, Sarah Post +1 more

Quasi-exactly solvable (QES) deformations of harmonic and singular oscillators have been widely studied via a range of approaches. In this paper, we obtain families of new QES deformations of solvable quantum systems associated with exceptional orthogonal polynomials (EOPs). The construction builds upon the theory of Darboux-Crum transformations and the classification of exactly solvable quantum systems associated with EOPs of Hermite type. It is shown that the deformations break the exact solvability of the undeformed systems and introduce model parameters into the deformed systems that permit the existence of a finite number of polynomial solutions whose roots satisfy systems of algebraic equations. The new families presented and studied consist of deformations of quantum systems related to Hermite EOPs of type III with arbitrary codimensions. We present polynomial and rational deformations and analyze, in each case, the conditions for quasi-exact solvability in terms of Bethe ansatz equations and parameter constraints. In general, the structures of the underlying polynomial solutions are no longer associated with well-known classical orthogonal polynomials. So for the general cases, we mainly focus on presenting the new approach as well as the Bethe ansatz equations and the constraints for model parameters. As applications, we construct particular families of QES deformations related to solvable models allowing one, two, and up to four gaps, and obtain the closed form expressions for their wavefunctions and spectra. We analyze the existence of QES solutions in the spaces of model parameters, providing information on the number of solutions for given parameters.

Sh. M. Nagiyev

We present two new limit relations that reduce the orthogonal pseudo-Jacobi polynomials directly to the Hermite polynomials with shifted and nonshifted arguments. The proofs of these limit relations are based on the method of mathematical induction. These limits open up the prospects for studying new exactly solvable harmonic oscillator models in homogeneous external fields in quantum mechanics in terms of pseudo-Jacobi polynomials. As an application of these limit relations, a model of a linear harmonic oscillator with a position-dependent mass in an external homogeneous gravitational field (a pseudo-Jacobi oscillator in an external field) is considered. The form of the generalized Hamiltonian for describing quantum mechanical systems with a position-dependent mass is presented.

Kumar Eswaran

Şirin A. Büyükaşık, Zehra Çayiç

We introduce exactly solvable quantum parametric oscillators, which are generalizations of the quantum problems related with the classical orthogonal polynomials of Hermite, Laguerre, and Jacobi type, introduced in the work of Büyükaşık et al. [J. Math. Phys. 50, 072102 (2009)]. Quantization of these models with specific damping, frequency, and external forces is obtained using the Wei-Norman Lie algebraic approach. This determines the evolution operator exactly in terms of two linearly independent homogeneous solutions and a particular solution of the corresponding classical equation of motion. Then, time-evolution of wave functions and coherent states are found explicitly. Probability densities, expectation values, and uncertainty relations are evaluated and their properties are investigated under the influence of the external terms.