On absolute continuity of the spectrum of a periodic magnetic Schr\"odinger operator

We consider the Schr\"odinger operator in ${\mathbb R}^n$, $n\geq 3$, with the electric potential $V$ and the magnetic potential $A$ being periodic functions (with a common period lattice) and prove absolute continuity of the spectrum of the operator in question under some conditions which, in particular, are satisfied if $V\in L^{n/2}_{{\mathrm {loc}}}({\mathbb R}^n)$ and $A\in H^q_{{\mathrm {loc}}}({\mathbb R}^n;{\mathbb R}^n)$, $q>(n-1)/2$.Comment: 25 page

Quasi-conformal mappings and periodic spectral problems in dimension two

No description supplie

JR.: On exponential representations of analytic functions in the upper half-plane with positive imaginary part

Abstract. We continue the study of boundary data maps, that is, generalizations of spectral parameter dependent Dirichlet-to-Neumann maps for (three-coefficient) Sturm-Liouville operators on the finite interval (a,b) , to more general boundary conditions, began in [8] and [17]. While these earlier studies of boundary data maps focused on the case of general separated boundary conditions at a and b , the present work develops a unified treatment for all possible self-adjoint boundary conditions (i.e., separated as well as non-separated ones). In the course of this paper we describe the connections with Krein's resolvent formula for self-adjoint extensions of the underlying minimal Sturm-Liouville operator (parametrized in terms of boundary conditions), with some emphasis on the Krein extension, develop the basic trace formulas for resolvent differences of self-adjoint extensions, especially, in terms of the associated spectral shift functions, and describe the connections between various parametrizations of all self-adjoint extensions, including the precise relation to von Neumann's basic parametrization in terms of unitary maps between deficiency subspaces. Mathematics subject classification (2010): Primary 34B05, 34B27, 34L40; Secondary 34B20, 34L05, 47A10, 47E05. Keywords and phrases: Self-adjoint Sturm-Liouville operators on a finite interval, boundary data maps, Krein-type resolvent formulas, spectral shift functions, perturbation determinants, parametrizations of self-adjoint extensions