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Continuum limit of resonances for discrete Schrödinger operators

Kentaro Kameoka, Shu Nakamura

Abstract

We consider complex resonances for discrete and continuous Schrödinger operators, and we show that the resonances of discrete models converge to resonances of continuous models in the continuum limit. The potential is supposed to be a sum of an exterior dilation analytic function and an exponentially decaying function, which may have local singularities. The proof employs a generalization of the norm resolvent convergence of discrete Schrödinger operators by Nakamura and Tadano (2021), combined with the complex distortion method in the Fourier space. Our results confirm that the complex resonances can be approximately computed using discrete Schrödinger operators. We also give a recipe for the construction of approximate discrete operators for Schrödinger operators with singular potentials.

Continuum limit of resonances for discrete Schrödinger operators

Abstract

We consider complex resonances for discrete and continuous Schrödinger operators, and we show that the resonances of discrete models converge to resonances of continuous models in the continuum limit. The potential is supposed to be a sum of an exterior dilation analytic function and an exponentially decaying function, which may have local singularities. The proof employs a generalization of the norm resolvent convergence of discrete Schrödinger operators by Nakamura and Tadano (2021), combined with the complex distortion method in the Fourier space. Our results confirm that the complex resonances can be approximately computed using discrete Schrödinger operators. We also give a recipe for the construction of approximate discrete operators for Schrödinger operators with singular potentials.

Paper Structure

This paper contains 12 sections, 13 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Continuous potentials
  3. Singular potentials
  4. Complex distortion in the Fourier space
  5. Fourier distortion for continuous Schrödinger operators
  6. Fourier distortion for discrete Schrödinger operators
  7. Continuum limit for continuous potentials
  8. Abstract scheme
  9. Embedding operator
  10. The kinetic term
  11. The potential term
  12. Continuum limit for singular potentials

Key Result

Theorem 1

Under Assumption asm-1, in $\Omega(\delta_0, c_0)$ including multiplicities.

Theorems & Definitions (32)

  • Remark 1.1
  • Theorem 1
  • Remark 1.2
  • Theorem 2
  • Theorem 3
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • ...and 22 more