Jussi Behrndt

Scattering matrices and Weyl functions

For a scattering system {A Θ , A 0 } consisting of self-adjoint extensions A Θ and A 0 of a symmetric operator A with finite deficiency indices, the scattering matrix {S Θ (λ)} and a spectral shift function ξ Θ are calculated in terms of the Weyl function associated with a boundary triplet for A*, and a simple proof of the Krein-Birman formula is given. The results are applied to singular Sturm-Liouville operators with scalar and matrix potentials, to Dirac operators and to Schrodinger operators with point interactions.

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.

Elliptic operators, Dirichlet-to-Neumann maps and quasi boundary triples

The notion of quasi boundary triples and their Weyl functions is reviewed and applied to self-adjointness and spectral problems for a class of elliptic, formally symmetric, second order partial differential expressions with variable coefficients on bounded domains.

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.

An Inverse Problem of Calderón Type with Partial Data

A generalized variant of the Calderón problem from electrical impedance tomography with partial data for anisotropic Lipschitz conductivities is considered in an arbitrary space dimension n ≥ 2. The following two results are shown: (i) The selfadjoint Dirichlet operator associated with an elliptic differential expression on a bounded Lipschitz domain is determined uniquely up to unitary equivalence by the knowledge of the Dirichlet-to-Neumann map on an open subset of the boundary, and (ii) the Dirichlet operator can be reconstructed from the residuals of the Dirichlet-to-Neumann map on this subset.

On the number of negative eigenvalues of the Laplacian on a metric graph

The number of negative eigenvalues of self-adjoint Laplacians on metric graphs is calculated in terms of the boundary conditions and the underlying geometric structure. This extends and complements earlier results by Kostrykin and Schrader (2006 Contemp. Math. 415 201-25).

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.

Spectral points of definite type and type π for linear operators and relations in Krein spaces

Spectral points of type …+ and type …i for closed linear operators and relations in Krein spaces are introduced with the help of approximative eigensequences. It turns out that these spectral points are stable under compact perturbations and perturbations small in the gap metric.

Non-real eigenvalues of singular indefinite Sturm-Liouville operators

We study a Sturm-Liouville expression with indefinite weight of the form sgn (−d^2/dx^2+V ) on \mathbb{R} and the non-real eigenvalues of an associated selfadjoint operator in a Krein space. For real-valued potentials V with a certain behaviour at \pm \infty we prove that there are no real eigenvalues and the number of non-real eigenvalues (counting multiplicities) coincides with the number of negative eigenvalues of the selfadjoint operator associated to −d^2/dx^2 + V in L^2(\mathbb{R}). The general results are illustrated with examples.

Accumulation of complex eigenvalues of indefinite Sturm?Liouville operators

Spectral properties of singular Sturm?Liouville operators of the form with the indefinite weight x sgn(x) on are studied. For a class of potentials with lim|x|??V(x) = 0 the accumulation of complex and real eigenvalues of A to zero is investigated and explicit eigenvalue problems are solved numerically.