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Discrete Schrödinger Operator on a Tree, Angelesco Potentials, and Their Perturbations

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Abstract

We consider a class of discrete Schrödinger operators on an infinite homogeneous rooted tree. Potentials for these operators are given by the coefficients of recurrence relations satisfied on a multidimensional lattice by multiple orthogonal polynomials. For operators on a binary tree with potentials generated by multiple orthogonal polynomials with respect to systems of measures supported on disjoint intervals (Angelesco systems) and for compact perturbations of such operators, we show that the essential spectrum is equal to the union of the intervals supporting the orthogonality measures.

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Funding

The work of the first author (Sections 1 and 2) was supported the Russian Science Foundation under grant 19-71-30004. The work of the second and third authors (Section 3) was supported by the Moscow Center for Fundamental and Applied Mathematics (project no. 20-03-01). The second author was also supported by the National Science Foundation (grants DMS-1464479 and DMS-1764245) and by the Van Vleck Professorship Research Award. The third author was also supported by the Simons Foundation (grant CGM-354538).

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Authors and Affiliations

  1. Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, Moscow, 125047, Russia

    A. I. Aptekarev, S. A. Denisov & M. L. Yattselev

  2. Department of Mathematics, University of Wisconsin–Madison, 480n Lincoln Dr., Madison, WI, 53706, USA

    S. A. Denisov

  3. Department of Mathematical Sciences, Indiana University–Purdue University Indianapolis, 402 N. Blackford Str., Indianapolis, IN, 46202, USA

    M. L. Yattselev

Authors
  1. A. I. Aptekarev
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  2. S. A. Denisov
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  3. M. L. Yattselev
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Correspondence to A. I. Aptekarev.

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Aptekarev, A.I., Denisov, S.A. & Yattselev, M.L. Discrete Schrödinger Operator on a Tree, Angelesco Potentials, and Their Perturbations. Proc. Steklov Inst. Math. 311, 1–9 (2020). https://doi.org/10.1134/S0081543820060012

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