Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators

Two point eigenvalue correlation for a class of non-selfadjoint operators under random perturbations

We consider a non-selfadjoint $h$-differential model operator $P_h$ in the semiclassical limit ($h\rightarrow 0$) subject to random perturbations with a small coupling constant $δ$. Assume that $\exp(-\frac{1}{Ch}) < δ\ll h^κ$ for constants $C,κ>0$ suitably large. Let $Σ$ be the closure of the range of the principal symbol. We study the $2$-point intensity measure of the random point process of eigenvalues of the randomly perturbed operator $P_h^δ$ and prove an $h$-asymptotic formula for the average $2$-point density of eigenvalues. With this we show that two eigenvalues of $P_h^δ$ in the inte…

On the spectral properties of non-selfadjoint discrete Schrödinger operators

Let $H_0$ be a purely absolutely continuous selfadjoint operator acting on some separable infinite-dimensional Hilbert space and $V$ be a compact non-selfadjoint perturbation. We relate the regularity properties of $V$ to various spectral properties of the perturbed operator $H_0+V$. The structure of the discrete spectrum and the embedded eigenvalues are analysed jointly with the existence of limiting absorption principles in a unified framework. Our results are based on a suitable combination of complex scaling techniques, resonance theory and positive commutators methods. Various results sca…

On nonsymmetric rank one singular perturbations of selfadjoint operators

We consider nonsymmetric rank one singular perturbations of a selfadjoint operator, i.e., an expression of the form $\tilde A = A + α\left\langle\cdot, ω_1\right\rangleω_2$, $ω_1\not = ω_2$, $α\in{\mathbb C}$, in a general case $ω_1,ω_2\in{\mathcal H}_{-2}$. Using a constructive description of the perturbed operator $\tilde A$, we investigate some spectral and approximations properties of $\tilde A$. The wave operators corresponding to the couple $A$, $\tilde A$ and a series of examples are also presented.

Eigenvalues of the bilayer graphene operator with a complex valued potential

We study the spectrum of a system of second order differential operator perturbed by a non-selfadjoint matrix valued potential. We prove that eigenvalues of the perturbed operator are located near the edges of the spectrum of the unperturbed operator.

Completeness and spectral synthesis of nonselfadjoint one-dimensional perturbations of selfadjoint operators

We study spectral properties of nonselfadjoint rank one perturbations of compact selfadjoint operators. The problems under consideration include completeness of eigenvectors, relations between completeness of the perturbed operator and its adjoint, and the spectral synthesis problem. We obtain new criteria for completeness and spectral synthesis in this class as well as a series of counterexamples which show that the spectral structure of rank one perturbations is, in general, unexpectedly rich and complicated. A parallel spectral theory is developed for one-dimensional singular perturbations …

Eigenvalue bounds for non-selfadjoint Dirac operators

In this work we prove that the eigenvalues of the $n$-dimensional massive Dirac operator $\mathscr{D}_0 + V$, $n\ge2$, perturbed by a possibly non-Hermitian potential $V$, are localized in the union of two disjoint disks of the complex plane, provided that $V$ is sufficiently small with respect to the mixed norms $L^1_{x_j} L^\infty_{\widehat{x}_j}$, for $j\in\{1,\dots,n\}$. In the massless case, we prove instead that the discrete spectrum is empty under the same smallness assumption on $V$, and in particular the spectrum is the same of the unperturbed operator, namely $σ(\mathscr{D}_0+V)=σ(\m…

Weyl law for semi-classical resonances with randomly perturbed potentials

In this work we consider semi-classical Schrödinger operators with potentials supported in a bounded strictly convex subset ${\cal O}$ of ${\bf R}^n$ with smooth boundary. Letting $h$ denote the semi-classical parameter, we consider certain classes of small random perturbations and show that with probability very close to 1, the number of resonances in rectangles $[a,b]-i[0,ch^{2/3}[$, is equal to the number of eigenvalues in $[a,b]$ of the Dirichlet realization of the unperturbed operator in ${\cal O}$ up to a small remainder.

Approximate solutions of inverse problems for nonlinear space fractional diffusion equations with randomly perturbed data

This paper is concerned with backward problem for nonlinear space fractional diffusion with additive noise on the right-hand side and the final value. To regularize the instable solution, we develop some new regularized method for solving the problem. In the case of constant coefficients, we use the truncation methods. In the case of perturbed time dependent coefficients, we apply a new quasi-reversibility method. We also show the convergence rate between the regularized solution and the sought solution under some a priori assumption on the sought solution.

Perturbations of Gibbs semigroups and the non-selfadjoint harmonic oscillator

Let $T$ be the generator of a $C_0$-semigroup $e^{-Tt}$ which is of finite trace for all $t>0$ (a Gibbs semigroup). Let $A$ be another closed operator, $T$-bounded with $T$-bound equal to zero. In general $T+A$ might not be the generator of a Gibbs semigroup. In the first half of this paper we give sufficient conditions on $A$ so that $T+A$ is the generator of a Gibbs semigroup. We determine these conditions in terms of the convergence of the Dyson-Phillips expansion corresponding to the perturbed semigroup in suitable Schatten-von Neumann norms. In the second half of the paper we consider $T=…

The Spectrum of Schrödinger Operators with Randomly Perturbed Ergodic Potentials

We consider Schrödinger operators in $\ell^2(\mathbb{Z})$ whose potentials are given by the sum of an ergodic term and a random term of Anderson type. Under the assumption that the ergodic term is generated by a homeomorphism of a connected compact metric space and a continuous sampling function, we show that the almost sure spectrum arises in an explicitly described way from the unperturbed spectrum and the topological support of the single-site distribution. In particular, assuming that the latter is compact and contains at least two points, this explicit description of the almost sure spect…

Variation of discrete spectra of non-negative operators in Krein spaces

We study the variation of the discrete spectrum of a bounded non-negative operator in a Krein space under a non-negative Schatten class perturbation of order $p$. It turns out that there exist so-called extended enumerations of discrete eigenvalues of the unperturbed and the perturbed operator, respectively, whose difference is an $\ell^p$-sequence. This result is a Krein space version of a theorem by T.Kato for bounded selfadjoint operators in Hilbert spaces.

Solvable models for the Schrodinger operators with $δ'$-like potentials

We turn back to the well known problem of interpretation of the Schrodinger operator with the pseudopotential being the first derivative of the Dirac function. We show that the problem in its conventional formulation contains hidden parameters and the choice of the proper selfadjoint operator is ambiguously determined. We study the asymptotic behavior of spectra and eigenvectors of the Hamiltonians with increasing smooth potentials perturbed by short-range potentials. Appropriate solvable models are constructed and the corresponding approximation theorems are proved. We introduce the concepts …

扰动Berezin-Toeplitz算子的概率Weyl律

A probabilistic Weyl-law for perturbed Berezin-Toeplitz operators

本文证明随机扰动Berezin-Toeplitz算子谱的概率Weyl律，推广Vogel 2020年结果，采用其策略及作者发展的奇异符号演算。

A random regularized approximate solution of the inverse problem for the Burgers' equation

In this paper, we find a regularized approximate solution for an inverse problem for the Burgers' equation. The solution of the inverse problem for the Burgers' equation is ill-posed, i.e., the solution does not depend continuously on the data. The approximate solution is the solution of a regularized equation with randomly perturbed coefficients and randomly perturbed final value and source functions. To find the regularized solution, we use the modified quasi-reversibility method associated with the truncated expansion method with nonparametric regression. We also investigate the convergence…

On Explicit Recursive Formulas in the Spectral Perturbation Analysis of a Jordan Block

Let A(z) be an analytic square matrix and $λ_{0}$ an eigenvalue of A(0) of multiplicity m. Then under the generic condition, the characteristic polynomial of A(z) evaluated at $λ_{0}$ has a simple zero at z=0, we prove that the Jordan normal form of A(0) corresponding to the eigenvalue $λ_{0}$ consists of a single m-by-m Jordan block, the perturbed eigenvalues near $λ_{0}$ and their eigenvectors can be represented by a single convergent Puiseux series containing only powers of z^{1/m}, and there are explicit recursive formulas to compute all the Puiseux series coefficients from just the deriva…

强残差收敛下非自伴随谱的尖锐上升-下降谱稳定性

Sharp Ascent--Descent Spectral Stability under Strong Resolvent Convergence

本文提出基于强残差收敛的非自伴随谱稳定性理论，通过计算有限元诊断γ_h保证谱稳定性，并验证其在计算实例中的有效性。

Wave operators, similarity and dynamics for a class of Schroedinger operators with generic non-mixed interface conditions in 1D

We consider a simple modification of the 1D-Laplacian where non-mixed interface conditions occur at the boundaries of a finite interval. It has recently been shown that Schrödinger operators having this form allow a new approach to the transverse quantum transport through resonant heterostructures. In this perspective, it is important to control the deformations effects introduced on the spectrum and on the time propagator by this class of non-selfadjont perturbations. In order to obtain uniform-in-time estimates of the perturbed semigroup, our strategy consists in constructing stationary wave…