Related papers: Compactness of Schr\"odinger semigroups
In this paper, we describe the leftmost eigenvalue of the non-selfadjoint operator $\mathcal{A}_h = -h^2\Delta+iV(x)$ with Dirichlet boundary conditions on a smooth bounded domain $\Omega\subset\mathbb{R}^n\,$, as $h\rightarrow0\,$. $V$ is…
We establish necessary and sufficient conditions for the boundedness of the relativistic Schr\"odinger operator $\mathcal{H} = \sqrt{-\Delta} + Q$ from the Sobolev space $W^{1/2}_2 (\R^n)$ to its dual $W^{-1/2}_2 (\R^n)$, for an arbitrary…
In this paper we obtain asymptotic formulas of arbitrary order for the Bloch eigenvalue and the Bloch function of the periodic Schrodinger operator of arbitrary dimension, when corresponding quasimomentum lies near a diffraction hyperplane.…
We investigate an extension of Schauder's theorem by studying the relationship between various $s$-numbers of an operator $T$ and its adjoint $T^*$. We have three main results. First, we present a new proof that the approximation number of…
This paper is devoted to self-adjoint cyclically compact operators on Hilbert--Kaplansky module over a ring of bounded measurable functions. The spectral theorem for such a class of operators are given. We apply this result to partial…
The spectral properties of two-dimensional Schr\"odinger operators with $\delta'$-potentials supported on star graphs are discussed. We describe the essential spectrum and give a complete description of situations in which the discrete…
We study spectral properties of one-dimensional singular perturbations of an unbounded selfadjoint operator and give criteria for the possibility to remove the whole spectrum by a perturbation of this type. A counterpart of our results for…
In this paper we are going to discuss compactness in Lorentz sequence spaces. Firstly, it will be shown how to define such a space, check whether a sequence belongs to it and calculate its norm. Equipped with this knowledge, we will proceed…
We study one-dimensional Schr\"odinger operators with complex measures as potentials and present an improved criterion for absence of eigenvalues which involves a weak local periodicity condition. The criterion leads to sharp quantitative…
We consider Schr\"odinger operators $H^h = (ih d+{\bf A})^* (ih d+{\bf A})$ with the periodic magnetic field ${\bf B}=d{\bf A}$ on covering spaces of compact manifolds. Under some assumptions on $\bf B$, we prove that there are arbitrarily…
We give general spectral and eigenvalue perturbation bounds for a selfadjoint operator perturbed in the sense of the pseudo-Friedrichs extension. We also give several generalisations of the aforementioned extension. The spectral bounds for…
We demonstrate criteria, purely based on finite subwords of the potential, to guarantee spectral inclusion as well as Hausdorff approximation of pseudospectra or even spectra of generalized Schr\"odinger operators on the discrete line or…
For a sequence of uniformly bounded, degenerate semigroups on a Hilbert space, we compare various types of convergences to a limit semigroup. Among others, we show that convergence of the semigroups, or of the resolvents of the generators,…
In this paper we characterize compactness of the canonical solution operator to d-bar on weigthed $L^2$ spaces on $\mathbb C.$ For this purpose we consider certain Schr\"odinger operators with magnetic fields and use a condition which is…
We study the compact perturbations of an isometry on a separable Hilbert space and provide a complete characterization of when they are quasinormal. Based on that, we present a complete classification for a rank-one perturbation of a…
In this study, we refine the compactification presented by Witz \cite{Witz} for general semigroups to the case of bounded $C_0$-semigroups, involving adjoint theory for this class of operators. This approach considerably reduces the…
We consider Schr\"odinger operators in $L^2(\mathbb{R}^3)$ with a singular interaction supported by a finite curve $\Gamma$. We present a proper definition of the operators and study their properties, in particular, we show that the…
We consider the Schr\"odinger operator $\mathcal L_{\alpha}$ 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…
We consider discrete one-dimensional Schr\"odinger operators with quasi-Sturmian potentials. We present a new approach to the trace map dynamical system which is independent of the initial conditions and establish a characterization of the…
We revisit the problem of semiclassical spectral asymptotics for a pure magnetic Schr\"odinger operator on a two-dimensional Riemannian manifold. We suppose that the minimal value $b_0$ of the intensity of the magnetic field is strictly…