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Semiclassical analysis and the Agmon-Finsler metric for discrete Schrödinger operators

Kentaro Kameoka

Abstract

The Agmon estimate for multi-dimensional discrete Schrödinger operators is studied with emphasis on the microlocal analysis on the torus. We first consider the semiclassical setting where semiclassical continuous Schrödinger operators are discretized with the mesh width proportional to the semiclassical parameter. Under this setting, the Agmon estimate for eigenfunctions is described by an Agmon metric, which is a Finsler metric rather than a Riemannian metric. Klein-Rosenberger (2008) proved this by a different argument in the case of a potential minimum. We also prove the Agmon estimate and the optimal anisotropic exponential decay of eigenfunctions for discrete Schrödinger operators in the non-semiclassical standard setting.

Semiclassical analysis and the Agmon-Finsler metric for discrete Schrödinger operators

Abstract

The Agmon estimate for multi-dimensional discrete Schrödinger operators is studied with emphasis on the microlocal analysis on the torus. We first consider the semiclassical setting where semiclassical continuous Schrödinger operators are discretized with the mesh width proportional to the semiclassical parameter. Under this setting, the Agmon estimate for eigenfunctions is described by an Agmon metric, which is a Finsler metric rather than a Riemannian metric. Klein-Rosenberger (2008) proved this by a different argument in the case of a potential minimum. We also prove the Agmon estimate and the optimal anisotropic exponential decay of eigenfunctions for discrete Schrödinger operators in the non-semiclassical standard setting.

Paper Structure

This paper contains 9 sections, 9 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Semiclassical Agmon estimate
  4. Calculation of exponentially conjugated operator
  5. The Agmon-Finsler metric
  6. Proof of Theorem \ref{['thm-2']}
  7. The exponential decay of eigenfunctions for discrete Schrödinger operators
  8. Proof of Theorem \ref{['thm-3']}
  9. The optimality of Theorem \ref{['thm-3']}

Key Result

Theorem 1

Under Assumption asm-Agmon and the above notation, for any $C_0>0$, $\delta_0>0$ and $\varepsilon>0$, there exist $C>0$, $h_0>0$, $0<\delta<\delta_0$, $\chi,\, \tilde{\chi}\in C^{\infty}_b(\mathbb{R}^d; [0, 1])$ with and $\rho \in C^{\infty}(\mathbb{R}^d; \mathbb{R}_{\ge 0})$ with such that for $0<h<h_0$, for any $u\in \ell^2(h\mathbb{Z}^d)$ and any $z\in [E-C_0, E+C_0 h]+i[-C_0, C_0]$.

Theorems & Definitions (23)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Remark 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.1
  • proof : Proof of Proposition \ref{['thm-1']}
  • Lemma 3.1
  • proof
  • ...and 13 more