Related papers: On a class of $J$-self-adjoint operators with empt…

We produce a new proof and extend results by Harrell and Stubbe for the discrete spectrum of a self-adjoint operator. An abstract approach--based on commutator algebra, the Rayleigh-Ritz principle, and an ``optimal'' usage of the…

The compression of the resolvent of a non-self-adjoint Schr\"odinger operator $-\Delta+V$ onto a subdomain $\Omega\subset\mathbb R^n$ is expressed in a Krein-Naimark type formula, where the Dirichlet realization on $\Omega$, the…

The famous results of M.G. Kre\u{\i}n concerning the description of selfadjoint contractive extensions of a Hermitian contraction $T_1$ and the characterization of all nonnegative selfadjoint extensions $\widetilde A$ of a nonnegative…

This is an expository paper about self-adjoint extensions of the Laplacian on R^d, initially defined on functions supported away from a point. Let L be the Laplacian with domain smooth functions with compact support away from the origin. We…

Let $A$ be a non-negative self-adjoint operator in a Hilbert space $\mathcal{H}$ and $A_{0}$ be some densely defined closed restriction of $A_{0}$, $A_{0}\subseteq A \neq A_{0}$. It is of interest to know whether $A$ is the unique…

In this note we investigate complete non-selfadjointness for all maximally dissipative extensions of a Schr\"odinger operator on a half-line with dissipative bounded potential and dissipative boundary condition. We show that all maximally…

We develop the concept of operators in Hilbert spaces which are similar to their adjoints via antiunitary operators, the latter being not necessarily involutive. We discuss extension theory, refined polar and singular-value decompositions,…

We give a self-contained presentation of the theory of self-adjoint extensions using the technique of boundary triples. A description of the spectra of self-adjoint extensions in terms of the corresponding Krein maps (Weyl functions) is…

We develop a general technique for finding self-adjoint extensions of a symmetric operator that respect a given set of its symmetries. Problems of this type naturally arise when considering two- and three-dimensional Schr\"odinger operators…

We study singular Schr\"odinger operators on a finite interval as selfadjoint extensions of a symmetric operator. We give sufficient conditions for the symmetric operator to be in the $n$-entire class, which was defined in our previous…

The question of self-adjoint realizations of sign-indefinite second order differential operators is discussed in terms of a model problem. Operators of the type $-\frac{d}{dx} \sgn (x) \frac{d}{dx}$ are generalized to finite, not…

Formally symmetric differential operators on weighted Hardy-Hilbert spaces are analyzed, along with adjoint pairs of differential operators. Eigenvalue problems for such operators are rather special, but include many of the classical…

The problem of construction a quantum mechanical evolution for the Schrodinger equation with a degenerate Hamiltonian which is a symmetric operator that does not have self-adjoint extensions is considered. Self-adjoint regularization of the…

We define Dirac operators on $\mathbb{S}^3$ (and $\mathbb{R}^3$) with magnetic fields supported on smooth, oriented links and prove self-adjointness of certain (natural) extensions. We then analyze their spectral properties and show, among…

We consider the two-dimensional Dirac operator with Lorentz-scalar $\delta$-shell interactions on each edge of a star-graph. An orthogonal decomposition is performed which shows such an operator is unitarily equivalent to an orthogonal sum…

We present a generic operator $J$ simply defined as a linear map not increasing the degree from the vectorial space of polynomial functions into itself and we address the problem of finding the polynomial sequences that coincide with the…

Given Hilbert space operators $T, S\in\B$, let $\triangle$ and $\delta\in B(\B)$ denote the elementary operators $\triangle_{T,S}(X)=(L_TR_S-I)(X)=TXS-X$ and $\delta_{T,S}(X)=(L_T-R_S)(X)=TX-XS$. Let $d=\triangle$ or $\delta$. Assuming $T$…

We find supersymmetric partners of a family of self-adjoint operators which are self-adjoint extensions of the differential operator $-d^2/dx^2$ on $L^2[-a,a]$, $a>0$, that is, the one dimensional infinite square well. First of all, we…

Theorem 1. Given a number c >= 1, there exists a J-unitary operator \hat{V}, such that: (a) r(\hat{V})= r(\hat{V}^{-1})= c ; (b) S(c^{-1}\hat{V})=S(c^{-1}\hat{V}^{-1}) =S(c^{-1}\hat{V}^{*-1}) = S(c^{-1}\hat{V}^*)={0} (c) there exist maximal…

We study two- and three-dimensional matrix Schr\"odinger operators with $m\in \mathbb N$ point interactions. Using the technique of boundary triplets and the corresponding Weyl functions, we complete and generalize the results obtained by…