Reconstruction of Differential Operators with Frozen Argument

We study spectral properties of a wide class of differential operators with frozen arguments by putting them into a general framework of rank-one perturbation theory. In particular, we give a complete characterization of possible eigenvalues for these operators and solve the inverse spectral problem of reconstructing the perturbation from the resulting spectrum. This approach provides a unified treatment of several recent studies and gives a clear explanation and interpretation of the obtained results.

SINGULARLY FINITE RANK NONSYMMETRIC PERTURBATIONS ${\mathcal H}_{-2}$-CLASS OF A SELF-ADJOINT OPERATOR

The singular nonsymmetric rank one perturbation of a self-adjoint operator from classes ${\mathcal H}_{-1}$ and ${\mathcal H}_{-2}$ was considered for the first time in works by Dudkin M.E. and Vdovenko T.I. \cite{k8,k9}. In the mentioned papers, some properties of the point spectrum are described, which occur during such perturbations. This paper proposes generalizations of the results presented in \cite{k8,k9} and \cite{k2} in the case of nonsymmetric class ${\mathcal H}_{-2}$ perturbations of finite rank. That is, the formal expression of the following is considered \begin{equation*} \tilde A=A+\sum \limits_{j=1}^{n}\alpha_j\langle\cdot,\omega_j\rangle\delta_j, \end{equation*} where $A$ is an unperturbed self-adjoint operator on a separable Hilbert space ${\mathcal H}$, $\alpha_j\in{\mathbb C}$, $\omega_j$, $\delta_j$, $j=1,2, ..., n<\infty$ are vectors from the negative space ${\mathcal H}_{-2}$ constructed by the operator $A$, $\langle\cdot,\cdot\rangle$ is the dual scalar product between positive and negative spaces.

Dimension of the Exceptional Set in the Aronszajn–Donoghue Theorem for Finite Rank Perturbations

The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $d$ perturbations $A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$, ${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$, with ${\operatorname{Ran}}{\textbf{B}}$ being cyclic for $A$, parametrized by $d\times d$ Hermitian matrices ${\boldsymbol{\alpha }}$, the singular parts of the spectral measures of $A$ and $A_{\boldsymbol{\alpha }}$ are mutually singular for all ${\boldsymbol{\alpha }}$ except for a small exceptional set $E$. It was shown earlier by the 1st two authors, see [4], that $E$ is a subset of measure zero of the space $\textbf{H}(d)$ of $d\times d$ Hermitian matrices. In this paper, we show that the set $E$ has small Hausdorff dimension, $\dim E \le \dim \textbf{H}(d)-1 = d^2-1$.

Application of a Rank-One Perturbation to Pendulum Systems

From a perturbation theory proposed by Mehl, et al., a study of the rank-one perturbation of the problems governed by pendulum systems is presented. Thus, a study of motion of the simple pendulum, double and triple pendulums with oscillating support, not coupled as coupled by a spring and double pendulum with fixed support is proposed. Finally (strong) stability and instability zones are calculated for each studied system.

On the spectrum of random anti-symmetric and tournament matrices

We consider a discrete, non-Hermitian random matrix model, which can be expressed as a shift of a rank-one perturbation of an anti-symmetric matrix. We show that, asymptotically almost surely, the real parts of the eigenvalues of the non-Hermitian matrix around any fixed index are interlaced with those of the anti-symmetric matrix. Along the way, we show that some tools recently developed to study the eigenvalue distributions of Hermitian matrices extend to the anti-symmetric setting.