Semiclassical shape resonances for magnetic Stark Hamiltonians

TL;DR

This paper investigates the properties of shape resonances in two-dimensional magnetic Stark Hamiltonians under semiclassical conditions, establishing a correspondence with eigenvalues of a reference operator.

Contribution

It proves a one-to-one correspondence between shape resonances and discrete eigenvalues for such Hamiltonians with potential wells, and derives related Weyl law and asymptotic behaviors.

Findings

Established a correspondence between shape resonances and eigenvalues.

Derived Weyl law for the number of resonances.

Analyzed asymptotic behavior of resonances near potential well bottoms.

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.