Vladimir Lotoreichik

Schrödinger Operators with δ and δ ′-Potentials Supported on Hypersurfaces

Self-adjoint Schrödinger operators with δ and δ′-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity results on the functions in their domains are obtained, and analogues of the Birman–Schwinger principle and a variant of Krein’s formula are shown. Furthermore, Schatten–von Neumann type estimates for the differences of the powers of the resolvents of the Schrödinger operators with δ and δ′-potentials, and the Schrödinger operator without a singular interaction are proved. An immediate consequence of these estimates is the existence and completeness of the wave operators of the corresponding scattering systems, as well as the unitary equivalence of the absolutely continuous parts of the singularly perturbed and unperturbed Schrödinger operators. In the proofs of our main theorems we make use of abstract methods from extension theory of symmetric operators, some algebraic considerations and results on elliptic regularity.

Schrödinger operators with δ- and δ′-interactions on Lipschitz surfaces and chromatic numbers of associated partitions

We investigate Schrodinger operators with δ- and δ′-interactions supported on hypersurfaces, which separate the Euclidean space into finitely many bounded and unbounded Lipschitz domains. It turns out that the combinatorial properties of the partition and the spectral properties of the corresponding operators are related. As the main result, we prove an operator inequality for the Schrodinger operators with δ- and δ′-interactions which is based on an optimal coloring and involves the chromatic number of the partition. This inequality implies various relations for the spectra of the Schrodinger operators and, in particular, it allows to transform known results for Schrodinger operators with δ-interactions to Schrodinger operators with δ′-interactions.

Schrödinger operators with δ-interactions supported on conical surfaces

We investigate the spectral properties of self-adjoint Schrodinger operators with attractive δ-interactions of constant strength α > 0 supported on conical surfaces in  3 . It is shown that the essential spectrum is given by α −+ ∞ [4 , ) 2 and that the discrete spectrum is infinite and accumulates to α − 4 2 . Furthermore, an asymptotic estimate of these eigenvalues is obtained.

Trace formulae and singular values of resolvent power differences of self-adjoint elliptic operators

In this note, self-adjoint realizations of second-order elliptic differential expressions with non-local Robin boundary conditions on a domain Ω ⊂ R n with smooth compact boundary are studied. A Schatten–von Neumann-type estimate for the singular values of the difference of the mth powers of the resolvents of two Robin realizations is obtained, and, for m>n /2 − 1, it is shown that the resolvent power difference is a trace class operator. The estimates are slightly stronger than the classical singular value estimates by Birman where one of the Robin realizations is replaced by the Dirichlet operator. In both cases, trace formulae are proved, in which the trace of the resolvent power differences in L 2 (Ω) is written in terms of the trace of derivatives of Neumann-to-Dirichlet and Robin-to-Neumann maps on the boundary space L 2 (∂Ω).

Approximation of Schrödinger operators with δ-interactions supported on hypersurfaces

We show that a Schrodinger operator Aδ,α with a δ-interaction of strength α supported on a bounded or unbounded C2-hypersurface Σ⊂Rd,d≥2, can be approximated in the norm resolvent sense by a family of Hamiltonians with suitably scaled regular potentials. The differential operator Aδ,α with a singular interaction is regarded as a self-adjoint realization of the formal differential expression −Δ−α⟨δΣ,·⟩δΣ, where α:Σ→R is an arbitrary bounded measurable function. We discuss also some spectral consequences of this approximation result.

A spectral isoperimetric inequality for cones

In this note, we investigate three-dimensional Schrödinger operators with

Quasi boundary triples and semibounded self-adjoint extensions

\delta

An Eigenvalue Inequality for Schrödinger Operators with δ- and δ’-interactions Supported on Hypersurfaces

δ-interactions supported on