Related papers: Continuity properties of Schr\"odinger semigroups …
By suitably extending a Feynman-Kac formula of Simon [Canadian Math. Soc. Conf. Proc, 28 (2000), 317-321], we study one-parameter semigroups generated by (the negative of) rather general Schroedinger operators, which may be unbounded from…
We extend the Feynman-Kac formula for Schr\"odinger type operators on vector bundles over noncompact Riemannian manifolds to possibly very singular potentials that appear in hydrogen like quantum mechanical problems and that need not be…
We construct multidimensional Schr\"odinger operators with a spectrum that has no gaps at high energies and that is nowhere dense at low energies. This gives the first example for which this widely expected topological structure of the…
This paper is concerned with emptyness of the essential spectrum, or equivalently compactness of the semigroup, for perturbations of selfadjoint operators that are bounded below (on an L^2-space). For perturbations by a (nonnegative)…
We analyze properties of semigroups generated by Schr\"odinger operators $-\Delta+V$ or polyharmonic operators $-(-\Delta)^m$, on metric graphs both on $L^p$-spaces and spaces of continuous functions. In the case of spatially constant…
We study Schr\"odinger operators on the real line whose potentials are generated by an underlying ergodic subshift over a finite alphabet and a rule that replaces symbols by compactly supported potential pieces. We first develop the…
We compare the bottom of the spectrum of discrete and continuous Schr\"odinger operators with periodic potentials with barriers at the boundaries of their fundamental domains. Our results show that these energy levels coincide in the…
We use the mirror coupling of Brownian motion to show that under a $\beta\in (0,1)$-dependent Kato type assumption (which is satisfied under a suitable $L^q$-assumption on the electro-magnetic potential, where $q$ depends on $\beta$ and the…
The object of the present study is the integrated density of states of a quantum particle in multi-dimensional Euclidean space which is characterized by a Schr{\"o}dinger operator with magnetic field and a random potential which may be…
With $(X,\mathfrak{d},\mathfrak{m})$ an $\mathrm{RCD}^*(K,N)$ space for some $K\in\mathbf{R}$, $N\in [1,\infty)$, let $H$ be the self-adjoint Laplacian induced by the underlying Cheeger form. Given $\alpha\in [0,1]$ we introduce the…
We define a Schr\"odinger operator on the half-space with a discontinuous magnetic field having a piecewise-constant strength and a uniform direction. Motivated by applications in the theory of superconductivity, we study the infimum of the…
We obtain several essential self-adjointness conditions for a Schroedinger type operator D*D+V acting in sections of a vector bundle over a manifold M. Here V is a locally square-integrable bundle map. Our conditions are expressed in terms…
We show that formal Schr\"odinger operators with singular potentials from the space W^{-1}_{2,unif}(R) can be naturally defined to give selfadjoint and bounded below operators, which depend continuously in the uniform resolvent sense on the…
We consider Schr\"odinger operators $H=-\Delta+V({\mathbf x})$ in ${\mathbb R}^d$, $d\geq2$, with quasi-periodic potentials $V({\mathbf x})$. We prove that the absolutely continuous spectrum of a generic $H$ contains a semi-axis…
We consider a Schr\"odinger operator $H=-\Delta+V(\vec x)$ in dimension two with a quasi-periodic potential $V(\vec x)$. We prove that the absolutely continuous spectrum of $H$ contains a semiaxis and there is a family of generalized…
This monograph develops the theory of covariant Schr\"odinger semigroups acting on sections of vector bundles over noncompact Riemannian manifolds from scratch. Contents: I. Sobolev spaces on vector bundles II. Smooth heat kernels on vector…
The two-dimensional Schroedinger operator with a uniform magnetic field and a periodic zero-range potential is considered. For weak magnetic fields we reduce the spectral problem to the semiclassical analysis of one-dimensional Harper-like…
For Schroedinger operators (including those with magnetic fields) with singular (locally integrable) scalar potentials on manifolds of bounded geometry, we study continuity properties of some related integral kernels: the heat kernel, the…
We study heat kernels of Schr\"odinger operators whose kinetic terms are non-local operators built for sufficiently regular symmetric L\'evy measures with radial decreasing profiles and potentials belong to Kato class. Our setting is fairly…
We define and examine certain matrix-valued multiplicative functionals with local Kato potential terms and use probabilistic techniques to prove that the semigroups of the corresponding partial differential operators with matrix-valued…