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arXiv · 2605.09812

Two-parameter classes of exactly solvable quantum systems

A. D. Alhaidari
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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.

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A. D. Alhaidari. 2026-05-10. Two-parameter classes of exactly solvable quantum systems. https://doi.org/10.1142/s0217751x2650154x

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