Related papers: Criticality theory for Schr\"odinger operators on …
We develop direct and inverse scattering theory for one-dimensional Schroedinger operators with steplike potentials which are asymptotically close to different finite-gap periodic potentials on different half-axes. We give a complete…
Differential operators of Schrodinger type are considered on metric Cayley graphs of free groups with a minimal set of M generators. The potential and edge lengths may vary with the M edge types. Using novel methods, a set of M multipliers…
We study the effect of non-negative potentials on the spectral gap of one-dimensional Schr\"odinger operators in the limit of large intervals. In particular, we derive upper and lower bounds on the gap for different classes of potentials…
Schrijver graphs are vertex-color-critical subgraphs of Kneser graphs having the same chromatic number. They also share the value of their fractional chromatic number but Schrijver graphs are not critical for that. Here we present an…
We derive inequalities for sums of eigenvalues of Schr\"{o}dinger operators on finite intervals and tori. In the first of these cases, the inequalities converge to the classical trace formulae in the limit as the number of eigenvalues…
We consider the Schr\"odinger operator on a star shaped graph with $n$ edges joined at a single vertex. We derive an expression for the trace of the difference of the perturbed and unperturbed resolvent in terms of a Wronskian. This leads…
For a large class of Schrodinger operators, we introduce the hyperbolic quadratic pencils by making the coupling constant dependent on the energy in the very special way. For these pencils, many problems of scattering theory are easier to…
We study Hardy inequalities for $p$-Schr\"odinger operators on general weighted graphs. Specifically, we prove a Maz'ya-type result, where we characterize the space of Hardy weights for $ p $-Schr\"odinger operators via a generalized…
Using the concept of intrinsic metric on a locally finite weighted graph, we give sufficient conditions for the magnetic Schr\"odinger operator to be essentially self-adjoint. The present paper is an extension of some recent results proven…
This paper deals with general structural properties of one-dimensional Schr"odinger operators with some absolutely continuous spectrum. The basic result says that the omega limit points of the potential under the shift map are…
We establish sharp two-sided weighted bounds on the fundamental solution to the fractional Schr\"{o}dinger operator using the method of desingularizing weights.
With appropriate notions of Hermitian vector bundles and connections over weighted graphs which we allow to be locally infinite, we prove Feynman-Kac-type representations for the corresponding semigroups and derive several applications…
We investigate the uniqueness, in suitable weighted $\ell^p$ spaces, of solutions to the Schr\"odinger equation with a potential, posed on infinite graphs. The potential can tend to zero at infinite with a certain rate.
We investigate trace formulas for one-dimensional Schroedinger operators which are trace class perturbations of quasi-periodic finite-gap operators using Krein's spectral shift theory. In particular, we establish the conserved quantities…
We consider a Sturm-Liouville operator a with integrable potential $q$ on the unit interval $I=[0,1]$. We consider a Schr\"odinger operator with a real compactly supported potential on the half line and on the line, where this potential…
We study Schr\"odinger operators on an infinite quantum graph of a chain form which consists of identical rings connected at the touching points by $\delta$-couplings with a parameter $\alpha\in\R$. If the graph is "straight", i.e. periodic…
We consider inverse dynamical, spectral, quantum and acoustical scattering problems for the Schr\"odinger operator on the half line. The goal of the paper is to establish the connections between different types of inverse data for these…
We develop the basic theory of ergodic Schr\"odinger operators, which is well known for ergodic probability measures, in the case of a base dynamics on an infinite measure space. This includes the almost sure constancy of the spectrum and…
Spectral properties of Schr\"odinger operators on compact metric graphs are studied and special emphasis is put on differences in the spectral behavior between different classes of vertex conditions. We survey recent results especially for…
In this paper, we discuss k-quasi-m-isometric composition operators and weighted composition operators on directed graphs with one circuit and more than one branching vertex.