Two-parameter classes of exactly solvable quantum systems
Abstract
We introduce two-parameter classes of exactly-solvable novel systems whose Hamiltonian operators could be represented by tridiagonal symmetric matrices in some orthogonal bases. 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. We obtain the associated two-parameter potential function induced by the change in the initial values that causes the system's wavefunction to change. 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.
Versions
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BibTeX
@misc{alhaidari2026twoparameter,
title = {{Two-parameter classes of exactly solvable quantum systems}},
author = {A. D. Alhaidari},
howpublished = {arXiv},
year = {2026},
month = may,
doi = {10.48550/arxiv.2605.09812},
eprint = {2605.09812},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.09812},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Two-parameter classes of exactly solvable quantum systems
AU - A. D. Alhaidari
PY - 2026
DA - 2026/05/27
PB - arXiv
DO - 10.48550/arxiv.2605.09812
UR - https://arxiv.org/abs/2605.09812
AB - We introduce two-parameter classes of exactly-solvable novel systems whose Hamiltonian operators could be represented by tridiagonal symmetric matrices in some orthogonal bases. 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. We obtain the associated two-parameter potential function induced by the change in the initial values that causes the system's wavefunction to change. 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.
ER -