A remark on imaginary part of resonance points

Some remarks on resonances in even-dimensional Euclidean scattering

The purpose of this paper is to prove some results about quantum mechanical black box scattering in even dimensions $d \geq 2$. We study the scattering matrix and prove some identities which hold for its meromorphic continuation onto $Λ$, the Riemann surface of the logarithm function. We relate the multiplicities of the poles of the continued scattering matrix to the multiplicities of the poles of the resolvent. Moreover, we show that the poles of the scattering matrix on the $m$th sheet of $Λ$ are related to the zeros of a scalar function defined on the physical sheet. This paper contains a n…

An example of resonance instability

We construct a semiclassical Schrödinger operator such that the imaginary part of its resonances closest to the real axis changes by a term of size $h$ when a real compactly supported potential of size $o ( h )$ is added.

二维Pauli算子在阈值处的共振

Resonances at the Threshold for Pauli Operators in Dimension Two

本文研究二维Pauli算子在阈值处零特征值受小参数正扰动后变为共振的现象，推导出共振虚部主项的显式表达式，揭示其虚部对参数的依赖由磁场通量决定，并分别处理非简并与简并情形。

Resonances in finitely perturbed quantum walks, resonance expansion and generic simplicity

We define resonances for finitely perturbed quantum walks as poles of the scattering matrix in the lower half plane. We show a resonance expansion which describes the time evolution in terms of resonances and corresponding Jordan chains. In particular, the decay rate of the survival probability is given by the imaginary part of resonances and the multiplicity. We prove generic simplicity of the resonances, although there are quantum walks with multiple resonances.

Spectral shift function and Resonances near the low ground state for Pauli and Schrödinger operators

We study the spectral shift function (SSF) $ξ(λ)$ and the resonances of the operator $H_V := \big( σ\cdot (-i\nabla - \textbf{A}) \big)^{2} + V$ in $L^2(\mathbb{R}^3)$ near the origin. Here $σ:= (σ_1,σ_2,σ_3)$ are the $2 \times 2$ Pauli matrices and $V$ is a hermitian potential decaying exponentially in the direction of the magnetic field $\textbf{B} := \text{curl} \hspace{0.6mm} \textbf{A}$. We give a representation of the derivative of the SSF as a sum of the imaginary part of a holomorphic function and a harmonic measure related to the resonances of $H_V$. This representation warrant the Br…

The Quantum Sabine Law for Resonances in Transmission Problems

We prove a quantum version of the Sabine law from acoustics describing the location of resonances in transmission problems. This work extends the author's previous work to a broader class of systems. Our main applications are to scattering by transparent obstacles, scattering by highly frequency dependent delta potentials, and boundary stabilized wave equations. We give a sharp characterization of the resonance free regions in terms of dynamical quantities. In particular, we relate the imaginary part of resonances or generalized eigenvalues to the chord lengths and reflectivity coefficients fo…