Two-parameter classes of exactly solvable quantum systems

TL;DR

This paper introduces two-parameter classes of exactly solvable quantum systems with Hamiltonians represented by tridiagonal matrices, featuring wavefunctions as series with orthogonal polynomial coefficients, and explores their spectral properties.

Contribution

The work develops a new class of exactly solvable quantum models characterized by two parameters, with wavefunctions expressed through orthogonal polynomials satisfying recursion relations.

Findings

Wavefunctions are series in basis elements with polynomial coefficients.

Bound states and resonances can emerge in continuous spectra when parameters exceed critical values.

Numerical realization of potentials is possible despite lack of analytical expressions.

Abstract

We introduce two-parameter classes of exactly-solvable novel systems whose Hamiltonian operators could be represented by tridiagonal symmetric matrices in some orthonormal basis set. The associated wavefunction is written as point-wise convergent series in the basis elements. The expansion coefficients of the series are orthogonal polynomials in the energy that satisfy the resulting three-term recursion relation starting with two-parameter initial values. These polynomials contain all physical information about the system and they depend on the values of the two parameters. However, we could not write down the associated two-parameter potential function analytically but could realize them numerically for a given set of physical parameters. We give several illustrative examples of these systems with continuous and/or discrete energy spectra. Moreover, a curious phenomenon is observed where bound states and/or resonances are induced in a system with pure continuous spectrum (e.g., a free particle) if the two parameters in the initial values exceed certain critical limits.