Abstract
We use the tridiagonal representation approach to enlarge the class of exactly solvable quantum systems. For this, we use a square-integrable basis in which the matrix representation of the wave operator is tridiagonal. In this case, the wave equation becomes a three-term recurrence relation for the expansion coefficients of the wave function with a solution in terms of orthogonal polynomials that is equivalent to a solution of the original problem. We obtain S-wave bound states for a new four-parameter potential with a 1/r2 singularity but short-range, which has an elaborate configuration structure and rich spectral properties. A particle scattered by this potential must overcome a barrier and can then be trapped in the potential valley in a resonance or bound state. Using complex rotation, we demonstrate the rich spectral properties of the potential in the case of a nonzero angular momentum and show how this structure varies with the parameters of the potential.
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Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 195, No. 3, pp. 422–436, June, 2018.
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Alhaidari, A.D. Four-Parameter 1/r2 Singular Short-Range Potential with Rich Bound States and A Resonance Spectrum. Theor Math Phys 195, 861–873 (2018). https://doi.org/10.1134/S0040577918060053
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DOI: https://doi.org/10.1134/S0040577918060053