This paper is concerned with emptyness of the essential spectrum, or equivalently compactness of the semigroup, for perturbations of selfadjoint operators that are bounded below (on an L2-space). For perturbations by a (nonnegative) potential we obtain a simple criterion for compactness of the semigroup in terms of relative compactness of the operators of multiplication with characteristic func...

Problems of existence of the singular spectrum on the continuous spectrum emerges in some mathematical aspects of quantum scattering theory and quantum solid physics. In the latter field, this phenomenon results from physical effects such as the Anderson transitions in dielectrics. In the study of this problem, selfadjoint Friedrichs model operators play an important part and constitute quite a...

We obtain new inequalities involving Berezin norm and number of bounded linear operators defined on a reproducing kernel Hilbert space $${\mathscr {H}}.$$ Among many obtained here, it is shown that if A positive operator {H}}$$ , then $$\Vert A\Vert _{ber}={{{\textbf {ber}}}}(A)$$ where _{ber}$$ $${{{\textbf are the A, respectively. In contrast to numerical radius, this equality does not hold f...

For a scattering system {AΘ, A0} consisting of selfadjoint extensions AΘ and A0 of a symmetric operator A with finite deficiency indices, the scattering matrix {SΘ(λ)} and a spectral shift function ξΘ are calculated in terms of the Weyl function associated with the boundary triplet for A∗ and a simple proof of the Krein-Birman formula is given. The results are applied to singular Sturm-Liouvill...

A reduction of the transmission eigenvalue problem for multiplicative sign-definite perturbations of elliptic operators with constant coefficients to an eigenvalue problem for a non-selfadjoint compact operator is given. Sufficient conditions for the existence of transmission eigenvalues and completeness of generalized eigenstates for the transmission eigenvalue problem are derived. In the trac...

In this paper we provide sharp error bounds in approximating the weighted Riemann-Stieltjes integral R b a f (t) g (t) d (t) by the weighted trapezoidal rule f(a)+f(b) 2 R b a g (t) d (t) : Applications for continuous functions of selfadjoint operators in complex Hilbert spaces are given as well.

For a scattering system {AΘ, A0} consisting of selfadjoint extensions AΘ and A0 of a symmetric operator A with finite deficiency indices, the scattering matrix {SΘ(λ)} and a spectral shift function ξΘ are calculated in terms of the Weyl function associated with the boundary triplet for A∗ and a simple proof of the Krein-Birman formula is given. The results are applied to singular Sturm-Liouvill...

A few numerical methods for linear evolution equations are developed and analyzed in this paper. These fourth order methods allow large step sizes for highly oscillatory equations when the evolution operators vary slowly in time. The methods are also conservative for equations such as the Schrr odinger equation, where the evolution operator is skew-selfadjoint.

Extension results, expressed in terms of complete boundedness, leading to necessary and sufficient conditions for the solvability of power moment problems with unbounded operator data are given. As an application, necessary and sufficient conditions for the existence of selfadjoint and normal extensions for some classes of commuting tuples of unbounded linear operators are obtained.