arXiv subjects

Vladimir Lotoreichik

Publications and source records attributed to Vladimir Lotoreichik.

77 records · Page 5Linked to original sources

arXiv2011

Schatten-von Neumann estimates for resolvent differences of Robin Laplacians on a half-space

The difference of the resolvents of two Laplacians on a half-space subject to Robin boundary conditions is studied. In general this difference is not compact, but it will be shown that it is compact and even belongs to some Schatten-von-Neumann class, if the coefficients in the boundary condition are sufficiently close to each other in a proper sense. In certain cases the resolvent difference is shown to belong even to the same Schatten-von Neumann class as it is known for the resolvent difference of two Robin Laplacians on a domain with a compact boundary.

math.SP↗

arXiv2010

Spectral estimates for resolvent differences of self-adjoint elliptic operators

The notion of quasi boundary triples and their Weyl functions is an abstract concept to treat spectral and boundary value problems for elliptic partial differential equations. In the present paper the abstract notion is further developed, and general theorems on resolvent differences belonging to operator ideals are proved. The results are applied to second order elliptic differential operators on bounded and exterior domains, and to partial differential operators with $δ$ and $δ'$-potentials supported on hypersurfaces.

math.SP↗

arXiv2010

Singular continuous spectrum of half-line Schrödinger operators with point interactions on a sparse set

We say that a discrete set $X =\{x_n\}_{n\in\dN_0}$ on the half-line $$0=x_0 < x_1 <x_2 <x_3<... <x_n<... <+\infty$$ is sparse if the distances $Δx_n = x_{n+1} -x_n$ between neighbouring points satisfy the condition $\frac{Δx_{n}}{Δx_{n-1}} \rightarrow +\infty$. In this paper half-line Schrödinger operators with point $δ$- and $δ^\prime$-interactions on a sparse set are considered. Assuming that strengths of point interactions tend to $\infty$ we give simple sufficient conditions for such Schrödinger operators to have non-empty singular continuous spectrum and to have purely singular continuous spectrum, which coincides with $\dR_+$.

math.SP↗

arXiv2009

A remark on Schatten-von Neumann properties of resolvent differences of generalized Robin Laplacians on bounded domains

In this note we investigate the asymptotic behaviour of the $s$-numbers of the resolvent difference of two generalized self-adjoint, maximal dissipative or maximal accumulative Robin Laplacians on a bounded domain $Ω$ with smooth boundary $\partialΩ$. For this we apply the recently introduced abstract notion of quasi boundary triples and Weyl functions from extension theory of symmetric operators together with Krein type resolvent formulae and well-known eigenvalue asymptotics of the Laplace-Beltrami operator on $\partialΩ$. It will be shown that the resolvent difference of two generalized Robin Laplacians belongs to the Schatten-von Neumann class of any order $p$ for which $p>(dimΩ-1)/3$. Moreover, we also give a simple sufficient condition for the resolvent difference of two generalized Robin Laplacians to belong to a Schatten-von Neumann class of arbitrary small order. Our results extend and complement classical theorems due to M.Sh.Birman on Schatten-von Neumann properties of the resolvent differences of Dirichlet, Neumann and self-adjoint Robin Laplacians.

math.SP↗

arXiv2009

Lower bound on the spectrum of the Schrödinger operator in the plane with delta-potential supported by a curve

We consider the Schrödinger operator in the plane with delta-potential supported by a curve. For the cases of an infinite curve and a finite loop we give estimates on the lower bound of the spectrum expressed explicitly through the strength of the interaction and a parameter which characterizes geometry of the curve. Going further we cut the curve into finite number of pieces and estimate the bottom of the spectrum using the parameters for the pieces. As an application of the elaborated theory we consider a curve with a finite number of cusps and "leaky" quantum graphs.

math.SP↗