Related papers: On the Spectra of Real and Complex Lam\'e Operator…
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…
We consider periodic Schr\"{o}dinger operators on the hexagonal lattice with self-adjoint vertex conditions that allow discontinuity and concentrated mass at the vertices. This model generalizes the periodic Schr\"{o}dinger operator on the…
We consider Schr\"odinger operator in dimension $d\ge 2$ with a singular interaction supported by an infinite family of concentric spheres, analogous to a system studied by Hempel and coauthors for regular potentials. The essential spectrum…
We consider discrete Schr\"odinger operators $H_{\mu Q}=\Delta+\mu Q$ with real periodic potentials $Q$ on periodic graphs, where $\Delta$ is the adjacency operator and $\mu\in\mathbb R$ is a coupling constant. The spectra of the operators…
We consider Schr\"odinger operators $H^h = (ih d+{\bf A})^* (ih d+{\bf A})$ with the periodic magnetic field ${\bf B}=d{\bf A}$ on covering spaces of compact manifolds. Under some assumptions on $\bf B$, we prove that there are arbitrarily…
We study a harmonic triangular lattice, which relaxes in the presence of a weak, short-wavelength periodic potential. Monte Carlo simulations reveal that the elastic lattice has only short-ranged positional correlations, despite the absence…
This paper is a contribution to harmonic analysis of compact solvmanifolds. We consider the four-dimensional oscillator group ${\rm Osc}_1$, which is a semi-direct product of the three-dimensional Heisenberg group and the real line. We…
We consider the Schr\"odinger operator $H = -\Delta + V$ in a layer or in a $d$-dimensional cylinder. The potential $V$ is assumed to be periodic with respect to some lattice. We establish the absolute continuity of $H$, assuming $V \in…
We consider the short-distance behaviour of the product of the Noether O(N) currents in the lattice nonlinear sigma-model. We compare the numerical results with the predictions of the operator product expansion, using one-loop perturbative…
We study the spectral flow of the Wilson-Dirac operator H(m) with and without an additional Sheikholeslami-Wohlert (SW) term on a variety of SU(3) lattice gauge field ensembles in the range $0\le m \le 2$. We have used ensembles generated…
We will study the spectral problem related to the Laplace operator in a singularly perturbed periodic waveguide. The waveguide is a quasi-cylinder with contains periodic arrangement of inclusions. On the boundary of the waveguide we…
We first develop a general framework for Laplace operators defined in terms of the combinatorial structure of a simplicial complex. This includes, among others, the graph Laplacian, the combinatorial Laplacian on simplicial complexes, the…
We consider the spin-$J$ XXZ-Hamiltonian on general graphs $\mathcal{G}$ and show its equivalence to a direct sum of discrete many-particle Schr\"odinger type operators on what we call "$N$-particle graphs with maximal local occupation…
This paper is devoted to providing quantitative bounds on the location of eigenvalues, both discrete and embedded, of non self-adjoint Lam\'e operators of elasticity $-\Delta^\ast + V$ in terms of suitable norms of the potential $V$. In…
We study the theory of scattering for a class of Hartree type equations with long range interactions in space dimension n > 2, including Hartree equations with potential V(x) = lambda |x|^{- gamma} with gamma < 1. For 1/2 < gamma < 1 we…
We investigate spectral properties of limit-periodic Schr\"odinger operators in $\ell^2(\Z)$. Our goal is to exhibit as rich a spectral picture as possible. We regard limit-periodic potentials as generated by continuous sampling along the…
We study the theory of scattering for a class of Hartree type equations with long range interactions in space dimension n > 2, including Hartree equations with potential V(x) = lambda |x|^{- gamma}. For 0 < gamma < or =1 we prove the…
We consider the Dirichlet Laplacian in a straight planar strip perturbed by a bounded periodic symmetric operator. We prove the classical Bethe-Sommerfeld conjecture for this operator, namely, that this operator has finitely many gaps in…
I will show that operator of analytic (harmonic) continuation on a lattice graph has a positive spectrum. I use a theorem about positivity of eigenvalues of totally positive matrices. I conjecture that by approximation the similar result…
Recently, Aizenman and Warzel discovered a mechanism for the appearance of absolutely continuous spectrum for random Schroedinger operators on the Bethe lattice through rare resonances (resonant delocalization). We extend their analysis to…