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Resonances and viscosity limit for the Wigner-von Neumann type Hamiltonian

Kentaro Kameoka, Shu Nakamura

Abstract

The resonances for the Wigner-von Neumann type Hamiltonian are defined by the periodic complex distortion in the Fourier space. Also, following Zworski, we characterize resonances as the limit points of discrete eigenvalues of the Hamiltonian with a quadratic complex absorbing potential in the viscosity type limit.

Resonances and viscosity limit for the Wigner-von Neumann type Hamiltonian

Abstract

The resonances for the Wigner-von Neumann type Hamiltonian are defined by the periodic complex distortion in the Fourier space. Also, following Zworski, we characterize resonances as the limit points of discrete eigenvalues of the Hamiltonian with a quadratic complex absorbing potential in the viscosity type limit.

Paper Structure

This paper contains 5 sections, 3 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Periodic distortion in the Fourier space
  3. Analyticity of $\widetilde{V}_{\theta}$
  4. Definition of resonances
  5. Viscosity limit

Key Result

Theorem 1.1

Under Assumption ass-1, there exists a complex neighborhood: $\Omega\subset \mathbb{C}$ of $[0, \infty)\setminus \mathcal{T}$ such that the following holds: For any $f, g \in L_{\mathrm{comp}}^{2}(\mathbb{R})$, the matrix element $(f, R_+(z)g)$ has a meromorphic continuation to $\Omega$.

Theorems & Definitions (14)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Definition 1
  • Remark 1.3
  • Theorem 1.2
  • Lemma 2.1
  • Remark 2.1
  • proof : Proof of Lemma \ref{['lem-1']}
  • proof : Proof of Theorem \ref{['thm-1']}
  • ...and 4 more