[Submitted on 10 May 2026 (v1), last revised 27 May 2026 (this version, v2)]

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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.

Submission history

From: A. D. Alhaidari [view email]
[v1] Sun, 10 May 2026 23:29:07 UTC (1,647 KB)
[v2] Wed, 27 May 2026 02:29:48 UTC (1,764 KB)

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