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Register a new userOn Symmetries in the Theory of Finite Rank Singular Perturbations
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For a nonnegative self-adjoint operator $A_0$ acting on a Hilbert space $mathfrak{H}$ singular perturbations of the form $A_0+V, V=sum_{1}^{n}{b}_{ij}<psi_j,cdot>psi_i$ are studied under some additional requirements of symmetry imposed on the initial operator $A_0$ and the singular elements $psi_j$. A concept of symmetry is defined by means of a one-parameter family of unitary operators $sU$ that is motivated by results due to R. S. Phillips. The abstract framework to study singular perturbations with symmetries developed in the paper allows one to incorporate physically meaningful connections between singular potentials $V$ and the corresponding self-adjoint realizations of $A_0+V$. The results are applied for the investigation of singular perturbations of the Schr{o}dinger operator in $L_2(dR^3)$ and for the study of a (fractional) textsf{p}-adic Schr{o}dinger type operator with point interactions.
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This article is dedicated to the following class of problems. Start with an $Ntimes N$ Hermitian matrix randomly picked from a matrix ensemble - the reference matrix. Applying a rank-$t$ perturbation to it, with $t$ taking the values $1le t le N$, we study the difference between the spectra of the perturbed and the reference matrices as a function of $t$ and its dependence on the underlying universality class of the random matrix ensemble. We consider both, the weaker kind of perturbation which either permutes or randomizes $t$ diagonal elements and a stronger perturbation randomizing successively $t$ rows and columns. In the first case we derive universal expressions in the scaled parameter $tau=t/N$ for the expectation of the variance of the spectral shift functions, choosing as random-matrix ensembles Dysons three Gaussian ensembles. In the second case we find an additional dependence on the matrix size $N$.
Bounds on the exponential decay of generalized eigenfunctions of bounded and unbounded selfadjoint Jacobi matrices are established. Two cases are considered separately: (i) the case in which the spectral parameter lies in a general gap of the spectrum of the Jacobi matrix and (ii) the case of a lower semi-bounded Jacobi matrix with values of the spectral parameter below the spectrum. It is demonstrated by examples that both results are sharp. We apply these results to obtain a many barriers-type criterion for the existence of square-summable generalized eigenfunctions of an unbounded Jacobi matrix at almost every value of the spectral parameter in suitable open sets. As an application, we provide examples of unbounded Jacobi matrices with a spectral mobility edge.
Let $A$ be a self-adjoint operator on a Hilbert space $fH$. Assume that the spectrum of $A$ consists of two disjoint components $sigma_0$ and $sigma_1$. Let $V$ be a bounded operator on $fH$, off-diagonal and $J$-self-adjoint with respect to the orthogonal decomposition $fH=fH_0oplusfH_1$ where $fH_0$ and $fH_1$ are the spectral subspaces of $A$ associated with the spectral sets $sigma_0$ and $sigma_1$, respectively. We find (optimal) conditions on $V$ guaranteeing that the perturbed operator $L=A+V$ is similar to a self-adjoint operator. Moreover, we prove a number of (sharp) norm bounds on variation of the spectral subspaces of $A$ under the perturbation $V$. Some of the results obtained are reformulated in terms of the Krein space theory. As an example, the quantum harmonic oscillator under a PT-symmetric perturbation is discussed.
We study perturbations of the self-adjoint periodic Sturm--Liouville operator [ A_0 = frac{1}{r_0}left(-frac{mathrm d}{mathrm dx} p_0 frac{mathrm d}{mathrm dx} + q_0right) ] and conclude under $L^1$-assumptions on the differences of the coefficients that the essential spectrum and absolutely continuous spectrum remain the same. If a finite first moment condition holds for the differences of the coefficients, then at most finitely many eigenvalues appear in the spectral gaps. This observation extends a seminal result by Rofe-Beketov from the 1960s. Finally, imposing a second moment condition we show that the band edges are no eigenvalues of the perturbed operator.
We consider metric perturbations of the Landau Hamiltonian. We investigate the asymptotic behaviour of the discrete spectrum of the perturbed operator near the Landau levels, for perturbations with power-like decay, exponential decay or compact support.
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