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Register a new userPushnitskis $mu$-invariant and Schrodinger operators with embedded eigenvalues
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Nurulla Azamov
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In this note, under a certain assumption on an affine space of operators, which admit embedded eigenvalues, it is shown that the singular part of the spectral shift function of any pair of operators from this space is an integer-valued function. The proof uses a natural decomposition of Pushnitskis $mu$-invariant into absolutely continuous and singular parts. As a corollary, the Birman-Krein formula follows.
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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 revisit an archive submission by P. B. Denton, S. J. Parke, T. Tao, and X. Zhang, arXiv:1908.03795, on $n times n$ self-adjoint matrices from the point of view of self-adjoint Dirichlet Schrodinger operators on a compact interval.
We consider a Schrodinger operator with complex-valued potentials on the line. The operator has essential spectrum on the half-line plus eigenvalues (counted with algebraic multiplicity) in the complex plane without the positive half-line. We determine series of trace formulas. Here we have the new term: a singular measure, which is absent for real potentials. Moreover, we estimate of sum of Im part of eigenvalues plus singular measure in terms of the norm of potentials. The proof is based on classical results about the Hardy spaces.
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.
In this article, we consider the semiclassical Schrodinger operator $P = - h^{2} Delta + V$ in $mathbb{R}^{d}$ with confining non-negative potential $V$ which vanishes, and study its low-lying eigenvalues $lambda_{k} ( P )$ as $h to 0$. First, we give a necessary and sufficient criterion upon $V^{-1} ( 0 )$ for $lambda_{1} ( P ) h^{- 2}$ to be bounded. When $d = 1$ and $V^{-1} ( 0 ) = { 0 }$, we are able to control the eigenvalues $lambda_{k} ( P )$ for monotonous potentials by a quantity linked to an interval $I_{h}$, determined by an implicit relation involving $V$ and $h$. Next, we consider the case where $V$ has a flat minimum, in the sense that it vanishes to infinite order. We give the asymptotic of the eigenvalues: they behave as the eigenvalues of the Dirichlet Laplacian on $I_{h}$. Our analysis includes an asymptotic of the associated eigenvectors and extends in particular cases to higher dimensions.
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