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Semiclassical shape resonances for magnetic Stark Hamiltonians

Kentaro Kameoka, Naoya Yoshida

Abstract

We study shape resonances of two-dimensional magnetic Stark Hamiltonians in the semiclassical limit. The magnetic field is assumed to be constant and the scalar potential is a perturbation of a linear potential. Under the assumption that the scalar potential has potential wells, the existence of a one-to-one correspondence between shape resonances of the Hamiltonian and discrete eigenvalues of a certain reference operator is proved. This implies the Weyl law for the number of resonances and the asymptotic behavior of the real parts of resonances near the bottom of a potential well. Resonances are studied as complex eigenvalues of complex distorted Hamiltonians, which is defined by the complex translation outside a compact set.

Semiclassical shape resonances for magnetic Stark Hamiltonians

Abstract

We study shape resonances of two-dimensional magnetic Stark Hamiltonians in the semiclassical limit. The magnetic field is assumed to be constant and the scalar potential is a perturbation of a linear potential. Under the assumption that the scalar potential has potential wells, the existence of a one-to-one correspondence between shape resonances of the Hamiltonian and discrete eigenvalues of a certain reference operator is proved. This implies the Weyl law for the number of resonances and the asymptotic behavior of the real parts of resonances near the bottom of a potential well. Resonances are studied as complex eigenvalues of complex distorted Hamiltonians, which is defined by the complex translation outside a compact set.

Paper Structure

This paper contains 7 sections, 13 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Exterior complex translation method for magnetic Stark Hamiltonians
  3. Definition and basic properties
  4. Proof of Lemma \ref{['lem-inv']}
  5. Semiclassical estimates of resonances
  6. Non-trapping estimate
  7. Shape resonance model

Key Result

Theorem 1

Under Assumption asm-res and Assumption asm-shape, there exists $S>0$ such that the numbers of resonances of $P(h)$ in $[a, b]-i[0, e^{-S/h}]$ satisfies when $h \to 0$.

Theorems & Definitions (27)

  • Theorem 1
  • Remark 1.1
  • Theorem 2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Lemma 2.3
  • proof : Proof of Proposition \ref{['prop2-2']} assuming Lemma \ref{['lem-inv']}
  • Proposition 2.4
  • proof
  • ...and 17 more