Related papers: The Schr\"odinger operator on graphs and topology
A family of discrete Schroedinger operators is investigated through scattering theory. The continuous spectrum of these operators exhibit changes of multiplicity, and some of these operators possess resonances at thresholds. It is shown…
Using an extension of the H\"ormander product of distributions, we obtain an intrinsic formulation of one-dimensional Schr\"odinger operators with singular potentials. This formulation is entirely defined in terms of standard {\it Schwartz}…
We consider discrete Schr\"odinger operators on the half line with potentials generated by the doubling map and continuous sampling functions. We show that the essential spectrum of these operators is always connected. This result is…
We study the uniqueness question for two inverse problems on graphs. Both problems consist in finding (possibly complex) edge or nodal based quantities from boundary measurements of solutions to the Dirichlet problem associated with a…
The scattering process on multiloop infinite p+1-valent graphs (generalized trees) is studied. These graphs are discrete spaces being quotients of the uniform tree over free acting discrete subgroups of the projective group $PGL(2, {\bf…
In this note we elaborate on the asymptotic behavior of the spectral gap of a class of discrete Schr\"odinger operators defined on a path graph in the limit of infinite volume. We confirm recent results and generalize them to a larger class…
We study singular Schr\"odinger operators on a finite interval as selfadjoint extensions of a symmetric operator. We give sufficient conditions for the symmetric operator to be in the $n$-entire class, which was defined in our previous…
We present the random behaviour of the Schr\"odinger map equation, a geometric partial differential equation, by considering its evolution for regular polygonal curves in both Euclidean and hyperbolic spaces. The results obtained are…
We consider the Schr\"odinger equation with a Hamiltonian given by a second order difference operator with nonconstant growing coefficients, on the half one dimensional lattice. This operator appeared first naturally in the construction and…
We discuss the behaviour of the bottom of the spectrum of scalar Schr\"odinger operators under Riemannian coverings of orbifolds. We apply our results to geometrically finite and to conformally compact orbifolds.
We study observable sets for Schr\"odinger equations on combinatorial graphs. For one-dimensional lattice Schr\"odinger operators \(H=-\Delta_{\mathrm{disc}}+V\) with \(V(n)\to c\in\mathbb R\) as \(|n|\to\infty\), we prove that a set…
We prove and apply two theorems: First, a quantitative, scale-free unique continuation estimate for functions in a spectral subspace of a Schr\"odinger operator on a bounded or unbounded domain, second, a perturbation and lifting estimate…
Some key features of the symmetries of the Schr\"odinger equation that are common to a much broader class of dynamical systems (some under construction) are illustrated. I discuss the algebra/superalgebra duality involving first and…
An explicit formula for the wave operators associated with Schroedinger operators on the discrete half-line is deduced from their stationary expressions. The formula enables us to understand the wave operators as one dimensional…
The singular real second order 1D Schrodinger operators are considered here with such potentials that all local solutions near singularities to the eigenvalue problem are meromorphic for all values of the spectral parameter. All…
We utilize the theory of de Branges spaces to show when certain Schr\"odinger operators with strongly singular potentials are uniquely determined by their associated spectral measure. The results are applied to obtain an inverse uniqueness…
We consider the spectral theory for discrete Schr\"odinger operators on the hexagonal lattice and their inverse scattering problem. We give a procedure for reconstructing the compactly supported potential from the scattering matrix for all…
Spectral components of one-dimensional Schr\"odinger operator with complex potential are investigated. An effective upper bound for the total number of eigenvalues and spectral singularities is established. For dissipative Schr\"odinger…
We study the influence of certain geometric perturbations on the spectra of self-adjoint Schr\"odinger operators on compact metric graphs. Results are obtained for permutation invariant vertex conditions, which, amongst others, include…
The matrix Schr\"odinger equation is considered on the half line with the general selfadjoint boundary condition at the origin described by two boundary matrices satisfying certain appropriate conditions. It is assumed that the matrix…