Hypergeometric Expression for the Resolvent of the Discrete Laplacian in Low Dimensions View Full Text


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Article Info

DATE

2021-06-04

AUTHORS

Kenichi Ito, Arne Jensen

ABSTRACT

We present an explicit formula for the resolvent of the discrete Laplacian on the square lattice, and compute its asymptotic expansions around thresholds in low dimensions. As a by-product we obtain a closed formula for the fundamental solution to the discrete Laplacian. For the proofs we express the resolvent in a general dimension in terms of the Appell–Lauricella hypergeometric function of type C outside a disk encircling the spectrum. In low dimensions it reduces to a generalized hypergeometric function, for which certain transformation formulas are available for the desired expansions. More... »

PAGES

32

References to SciGraph publications

  • 2011-06-18. Spectral estimates for Schrödinger operators with sparse potentials on graphs in JOURNAL OF MATHEMATICAL SCIENCES
  • 2011-07-29. 70+ Years of the Watson Integrals in JOURNAL OF STATISTICAL PHYSICS
  • 2018-09-11. Exterior diffraction problems for two-dimensional square lattice in ZEITSCHRIFT FÜR ANGEWANDTE MATHEMATIK UND PHYSIK
  • 2011-11-05. Inverse Problems, Trace Formulae for Discrete Schrödinger Operators in ANNALES HENRI POINCARÉ
  • 2020-05-05. Some Properties of Threshold Eigenstates and Resonant States of Discrete Schrödinger Operators in ANNALES HENRI POINCARÉ
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    http://scigraph.springernature.com/pub.10.1007/s00020-021-02648-2

    DOI

    http://dx.doi.org/10.1007/s00020-021-02648-2

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