Nonrelativistic Limit of Generalized MIT Bag Models and Spectral Inequalities.

For a family of self-adjoint Dirac operators - i c ( α · ∇ ) + c 2 2 subject to generalized MIT bag boundary conditions on domains in R 3 , it is shown that the nonrelativistic limit in the norm resolvent sense is the Dirichlet Laplacian. This allows to transfer spectral geometry results for Dirichlet Laplacians to Dirac operators for large c.

Schrödinger Operators with Oblique Transmission Conditions in R2.

In this paper we study the spectrum of self-adjoint Schrödinger operators in L2(R2) with a new type of transmission conditions along a smooth closed curve Σ⊆R2. Although these oblique transmission conditions are formally similar to δ'-conditions on Σ (instead of the normal derivative here the Wirtinger derivative is used) the spectral properties are significantly different: it turns out that for attractive interaction strengths the discrete spectrum is always unbounded from below. Besides this unexpected spectral effect we also identify the essential spectrum, and we prove a Krein-type resolvent formula and a Birman-Schwinger principle. Furthermore, we show that these Schrödinger operators with oblique transmission conditions arise naturally as non-relativistic limits of Dirac operators with electrostatic and Lorentz scalar δ-interactions justifying their usage as models in quantum mechanics.

Spectral Transition for Dirac Operators with Electrostatic δ -Shell Potentials Supported on the Straight Line.

In this note the two dimensional Dirac operator A η with an electrostatic δ -shell interaction of strength η ∈ R supported on a straight line is studied. We observe a spectral transition in the sense that for the critical interaction strengths η = ± 2 the continuous spectrum of A η inside the spectral gap of the free Dirac operator A 0 collapses abruptly to a single point.

A unified approach to Schrödinger evolution of superoscillations and supershifts.

Superoscillating functions and supershifts appear naturally in weak measurements in physics. Their evolution as initial conditions in the time-dependent Schrödinger equation is an important and challenging problem in quantum mechanics and mathematical analysis. The concept that encodes the persistence of superoscillations during the evolution is the (more general) supershift property of the solution. In this paper, we give a unified approach to determine the supershift property for the solution of the time-dependent one-dimensional Schrödinger equation. The main advantage and novelty of our results is that they only require suitable estimates and regularity assumptions on the Green's function, but not its explicit form. With this efficient general technique, we are able to treat various potentials.

Self-Adjoint Dirac Operators on Domains in R 3.

In this paper, the spectral and scattering properties of a family of self-adjoint Dirac operators in L 2 ( Ω ; C 4 ) , where Ω âŠ‚ R 3 is either a bounded or an unbounded domain with a compact C 2 -smooth boundary, are studied in a systematic way. These operators can be viewed as the natural relativistic counterpart of Laplacians with boundary conditions as of Robin type. Our approach is based on abstract boundary triple techniques from extension theory of symmetric operators and a thorough study of certain classes of (boundary) integral operators, that appear in a Krein-type resolvent formula. The analysis of the perturbation term in this formula leads to a description of the spectrum and a Birman-Schwinger principle, a qualitative understanding of the scattering properties in the case that Ω is an exterior domain, and corresponding trace formulas.

Spectral shift functions and Dirichlet-to-Neumann maps.

The spectral shift function of a pair of self-adjoint operators is expressed via an abstract operator-valued Titchmarsh-Weyl m-function. This general result is applied to different self-adjoint realizations of second-order elliptic partial differential operators on smooth domains with compact boundaries and Schrödinger operators with compactly supported potentials. In these applications the spectral shift function is determined in an explicit form with the help of (energy parameter dependent) Dirichlet-to-Neumann maps.

Elliptic differential operators on Lipschitz domains and abstract boundary value problems.

This paper consists of two parts. In the first part, which is of more abstract nature, the notion of quasi-boundary triples and associated Weyl functions is developed further in such a way that it can be applied to elliptic boundary value problems on non-smooth domains. A key feature is the extension of the boundary maps by continuity to the duals of certain range spaces, which directly leads to a description of all self-adjoint extensions of the underlying symmetric operator with the help of abstract boundary values. In the second part of the paper a complete description is obtained of all self-adjoint realizations of the Laplacian on bounded Lipschitz domains, as well as Krein type resolvent formulas and a spectral characterization in terms of energy dependent Dirichlet-to-Neumann maps. These results can be viewed as the natural generalization of recent results by Gesztesy and Mitrea for quasi-convex domains. In this connection we also characterize the maximal range spaces of the Dirichlet and Neumann trace operators on a bounded Lipschitz domain in terms of the Dirichlet-to-Neumann map. The general results from the first part of the paper are also applied to higher order elliptic operators on smooth domains, and particular attention is paid to the second order case which is illustrated with various examples.