Stability of Singular Spectral Types under Decaying Perturbations

On the Spectral Properties of the Landau Hamiltonian Perturbed by a moderately Decaying Magnetic Field

The Landau Hamiltonian, describing the behavior of a quantum particle in dimension 2 in a constant magnetic field, is perturbed by a magnetic field with power-like decay at infinity and a similar electric potential. We describe how the spectral subspaces change and how the Landau levels split under this perturbation.

Sine kernel asymptotics for a class of singular measures

We construct a family of measures on $\bbR$ that are purely singular with respect to Lebesgue measure, and yet exhibit universal sine-kernel asymptotics in the bulk. The measures are best described via their Jacobi recursion coefficients: these are sparse perturbations of the recursion coefficients corresponding to Chebyshev polynomials of the second kind. We prove convergence of the renormalized Christoffel-Darboux kernel to the sine kernel for any sufficiently sparse decaying perturbation.

Meromorphic continuations of finite gap Herglotz functions and periodic Jacobi matrices

We find a necessary and sufficient condition for a Herglotz function $m$ to be the Borel transform of the spectral measure of an exponentially decaying perturbation of a periodic Jacobi matrix. The condition is in terms of meromorphic continuation of $m$ to a natural Riemann surface and the structure of its zeros and poles. The analogous result is also established for the Borel transform of the spectral measure of eventually periodic Jacobi matrices. This paper generalizes the corresponding result from [17] for exponentially decaying perturbations of the free Jacobi matrix.

Preservation of the absolutely continuous spectrum of Schrödinger equation under perturbations by slowly decreasing potentials and a.e. convergence of integral operators

We prove new criteria of stability of the absolutely continuous spectrum of one-dimensional Schrödinger operators under slowly decaying perturbations. As applications, we show that the absolutely continuous spectrum of the free and periodic Schrödinger operators is preserved under perturbations by all potentials $V(x)$ satisfying $|V(x)| \leq C(1+x)^{-\frac{2}{3}-ε}.$ The main new technique includes an a.e.\ convergence theorem for a class of integral operators.

On Symmetries in the Theory of Finite Rank Singular Perturbations

For a nonnegative self-adjoint operator $A_0$ acting on a Hilbert space $\mathfrak{H}$ singular perturbations of the form $A_0+V, \ V=\sum_{1}^{n}{b}_{ij}<ψ_j,\cdot>ψ_i$ are studied under some additional requirements of symmetry imposed on the initial operator $A_0$ and the singular elements $ψ_j$. 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…

衰减势能的特征值间距上界

Upper bounds on eigenvalue spacing for decaying potentials

本文研究了衰减势能下Dirichlet特征值间距的上界，揭示了势能衰减率与特征值间距的关系。

Spectral and resonance problem for perturbations of periodic Jacobi operators

Necessary and sufficient conditions are presented for a measure to be the spectral measure of a finite range or exponentially decaying perturbation of a periodic Jacobi operator. As a corollary we can fully solve the inverse resonance problem: given resonances and eigenvalues we can recover the spectral measure of the Jacobi operator; we provide necessary and sufficient conditions under which such an operator exists and is unique; and we show that the inverse resonance problem is stable under small perturbations.

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.

Selfadjoint extensions of the multiplication operator in de Branges spaces as singular rank-one perturbations

We derive a description of the family of canonical selfadjoint extensions of the operator of multiplication in a de Branges space in terms of singular rank-one perturbations using distinguished elements from the set of functions associated with a de Branges space. The scale of rigged Hilbert spaces associated with this construction is also studied from the viewpoint of de Branges's theory.

Eigenvalue asymptotics of perturbed periodic Dirac systems in the slow-decay limit

A perturbation decaying to 0 at infinity and not too irregular at 0 introduces at most a discrete set of eigenvalues into the spectral gaps of a one-dimensional Dirac operator on the half-line. We show that the number of these eigenvalues in a compact subset of a gap in the essential spectrum is given by a quasi-semiclassical asymptotic formula in the slow-decay limit, which for power-decaying perturbations is equivalent to the large-coupling limit. This asymptotic behaviour elucidates the origin of the dense point spectrum observed in spherically symmetric, radially periodic three-dimensional…

A note on a conjecture concerning rank one perturbations of singular M-matrices

A conjecture from a paper by J. Bierkens and A.C.M. Ran concerning the location of eigenvalues of rank one perturbations of singular M-matrices is shown to be false in dimension four and higher, but true for dimension two, as well as for dimension three with an additional condition on the perturbation.

Zeroes of the spectral density of the Schroedinger operator with the slowly decaying Wigner-von Neumann potential

We consider the Schrödinger operator $\mathcal L_α$ on the half-line with a periodic background potential and a perturbation which consists of two parts: a summable potential and the slowly decaying Wigner--von Neumann potential $\frac{c\sin(2ωx+δ)}{x^γ}$, where $γ\in(\frac12,1)$. The continuous spectrum of this operator has the same band-gap structure as the continuous spectrum of the unperturbed periodic operator. In every band there exist two points, called critical, where the eigenfunction equation has square summable solutions. Every critical point $ν_{cr}$ is an eigenvalue of the operato…

Correct Singular Perturbations of the Laplace Operator with the Spectrum of the Unperturbed Operator

The work is devoted to the study of Laplace operator when the potential is a singular generalized function and plays the role of a singular perturbation of a Laplace operator. Abstract theorem obtained earlier by the authors B.N. Biyarov and G.K. Abdrasheva applies to this. The main purpose of the study is the spectral issue. Singular perturbations for differential operators have been studied by many authors for the mathematical substantiation of solvable models of quantum mechanics, atomic physics, and solid state physics. In all these cases, the problems were self-adjoint. In this paper, we …