Abstract
Using the factorization method, we construct finite-difference Schrödinger operators (Jacobi matrices) whose discrete spectra are composed from independent arithmetic, or geometric series. Such systems originate from the periodic, orq-periodic closure of a chain of corresponding Darboux transformations. The Charlier, Krawtchouk, Meixner orthogonal polynomials, theirq-analogs, and some other classical polynomials appear as the simplest examples forN = 1 andN = 2 (N is the period of closure). A natural generalization involves discrete versions of the Painlevé transcendents.
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On leave of absence from the Institute for Nuclear Research, Russian Academy of Sciences, Moscow, Russia.
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Spiridonov, V., Vinet, L. & Zhedanov, A. Difference schrödinger operators with linear and exponential discrete spectra. Lett Math Phys 29, 63–73 (1993). https://doi.org/10.1007/BF00760860
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DOI: https://doi.org/10.1007/BF00760860