Absence of embedded eigenvalues for Hamiltonian with crossed magnetic and electric fields

On the virial theorem for the relativistic operator of Brown and Ravenhall, and the absence of embedded eigenvalues

A virial theorem is established for the operator proposed by Brown and Ravenhall as a model for relativistic one-electron atoms. As a consequence, it is proved that the operator has no eigenvalues greater than $\max(m c^2, 2 αZ - \frac{1}{2})$, where $α$ is the fine structure constant, for all values of the nuclear charge $Z$ below the critical value $Z_c$: in particular there are no eigenvalues embedded in the essential spectrum when $Z \leq 3/4 α$. Implications for the operators in the partial wave decomposition are also described.

关于周期格点上离散薛定谔算子在连续谱中无嵌入特征值的一个注记

A remark on the absence of eigenvalues in continuous spectra for discrete Schrödinger operators on periodic lattices

本文在包含正方、三角、六角格点及其梯状结构的一类周期格点上，对具有指数衰减位势的离散薛定谔算子证明 Rellich-Vekua 型定理，并由此讨论唯一延拓性与连续谱中无嵌入特征值。

Resolvent expansions and continuity of the scattering matrix at embedded thresholds: the case of quantum waveguides

We present an inversion formula which can be used to obtain resolvent expansions near embedded thresholds. As an application, we prove for a class of quantum waveguides the absence of accumulation of eigenvalues and the continuity of the scattering matrix at all thresholds.

电磁狄拉克算子的点谱

On the point spectrum of electromagnetic Dirac operators

本文提出了一种用于电磁狄拉克算子的方法，以确定点谱和嵌入本征值的不存在条件。

Estimates on complex eigenvalues for Dirac operators on the half-line

We derive bounds on the location of non-embedded eigenvalues of Dirac operators on the half-line with non-Hermitian $L^1$-potentials. The results are sharp in the non-relativistic or weak-coupling limit. In the massless case, the absence of discrete spectrum is proved under a smallness assumption.

Embedded Eigenvalues and the Nonlinear Schrodinger Equation

A common challenge to proving asymptotic stability of solitary waves is understanding the spectrum of the operator associated with the linearized flow. The existence of eigenvalues can inhibit the dispersive estimates key to proving stability. Following the work of Marzuola & Simpson, we prove the absence of embedded eigenvalues for a collection of nonlinear Schrodinger equations, including some one and three dimensional supercritical equations, and the three dimensional cubic-quintic equation. Our results also rule out nonzero eigenvalues within the spectral gap and, in 3D, endpoint resonance…

Schrodinger operators with slowly decaying Wigner--von Neumann type potentials

We consider Schrödinger operators with potentials satisfying a generalized bounded variation condition at infinity and an $L^p$ decay condition. This class of potentials includes slowly decaying Wigner--von Neumann type potentials $\sin(ax)/x^b$ with $b>0$. We prove absence of singular continuous spectrum and show that embedded eigenvalues in the continuous spectrum can only take values from an explicit finite set. Conversely, we construct examples where such embedded eigenvalues are present, with exact asymptotics for the corresponding eigensolutions.

Eigenvalue estimates for non-selfadjoint Dirac operators on the real line

We show that the non-embedded eigenvalues of the Dirac operator on the real line with non-Hermitian potential $V$ lie in the disjoint union of two disks in the right and left half plane, respectively, provided that the $L^1-norm$ of $V$ is bounded from above by the speed of light times the reduced Planck constant. An analogous result for the Schrödinger operator, originally proved by Abramov, Aslanyan and Davies, emerges in the nonrelativistic limit. For massless Dirac operators, the condition on $V$ implies the absence of nonreal eigenvalues. Our results are further generalized to potentials …