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Register a new userOn Absence of Threshold Resonances for Schrodinger and Dirac Operators
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Using a unified approach employing a homogeneous Lippmann-Schwinger-type equation satisfied by resonance functions and basic facts on Riesz potentials, we discuss the absence of threshold resonances for Dirac and Schrodinger operators with sufficiently short-range interactions in general space dimensions. More specifically, assuming a sufficient power law decay of potentials, we derive the absence of zero-energy resonances for massless Dirac operators in space dimensions $n geq 3$, the absence of resonances at $pm m$ for massive Dirac operators (with mass $m > 0$) in dimensions $n geq 5$, and recall the well-known case of absence of zero-energy resonances for Schrodinger operators in dimension $n geq 5$.
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Let $Lambdasubset mathbb{R}^d$ be a domain consisting of several cylinders attached to a bounded center. One says that $Lambda$ admits a threshold resonance if there exists a non-trivial bounded function $u$ solving $-Delta u= u u$ in $Lambda$ and vanishing at the boundary, where $ u$ is the bottom of the essential spectrum of the Dirichlet Laplacian in $Lambda$. We derive a sufficient condition for the absence of threshold resonances in terms of the Laplacian eigenvalues on the center. The proof is elementary and is based on the min-max principle. Some two- and three-dimensional examples and applications to the study of Laplacians on thin networks are discussed.
We consider non-local Schrodinger operators with kinetic terms given by several different types of functions of the Laplacian and potentials decaying to zero at infinity, and derive conditions ruling embedded eigenvalues out. These results contrast and complement recent work on showing the existence of such eigenvalues occurring for the same types of operators under different conditions. Our goal in this paper is to advance techniques based on virial theorems, Mourre estimates, and an extended version of the Birman-Schwinger principle, previously developed for classical Schrodinger operators but thus far not used for non-local operators. We also present a number of specific cases by choosing particular classes of kinetic and potential terms of immediate interest.
We show that a generic quasi-periodic Schrodinger operator in $L^2(mathbb{R})$ has purely singular spectrum. That is, for any minimal translation flow on a finite-dimensional torus, there is a residual set of continuous sampling functions such that for each of these sampling functions, the Schrodinger operator with the resulting potential has empty absolutely continuous spectrum.
The present work aims at obtaining estimates for transformation operators for one-dimensional perturbed radial Schrodinger operators. It provides more details and suitable extensions to already existing results, that are needed in other recent contributions dealing with these kinds of operators.
We show that the spectral flow of a one-parameter family of Schrodinger operators on a metric graph is equal to the Maslov index of a path of Lagrangian subspaces describing the vertex conditions. In addition, we derive an Hadamard-type formula for the derivatives of the eigenvalue curves via the Maslov crossing form.
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