Related papers: Level repulsion for Schroedinger operators with si…
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…
Is considered the asymptotical behavior of spectral function $\rho(\lambda, \epsilon),\epsilon > 0$, of one family of self adjoint differential operators of second order, defined in space $L_2[0,+\infty)$ with potentials, depending on…
We provide a concise, yet fairly complete discussion of the concept of essential closures of subsets of the real axis and their intimate connection with the topological support of absolutely continuous measures. As an elementary application…
Level repulsion is associated with exceptional points which are square root singularities of the energies as functions of a (complex) interaction parameter. This is also valid for resonance state energies. Using this concept it is argued…
It is known that the spectrum of Schr\"odinger operators with sparse potentials consists of singular continuous spectrum. We give a sufficient condition so that the edge of the singular continuous spectrum is not an eigenvalue and construct…
In this paper we study the spectrum of self-adjoint Schr\"odinger operators in $L^2(\mathbb{R}^2)$ with a new type of transmission conditions along a smooth closed curve $\Sigma\subseteq \mathbb{R}^2$. Although these $\textit{oblique}$…
We establish connections between Schur parameters of the Schur class operator-valued functions, the corresponding simple conservative realizations, lower triangular Toeplitz matrices, and Kre\u\i n shorted operators. By means of Schur…
We study the level statistics for two classes of 1-dimensional random Schr\"odinger operators : (1) for operators whose coupling constants decay as the system size becomes large, and (2) for operators with critically decaying random…
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…
We study the spectra for a class of differential operators with asymptotically constant coefficients.These operators widely arise as the linearizations of nonlinear partial differential equations about patterns or nonlinear waves. We…
We study the semi-classical trace formula at a critical energy level for a Schr\"odinger operator on $\mathbb{R}^{n}$. We assume here that the potential has a totally degenerate critical point associated to a local maximum. The main result,…
We study spectral properties of Schr\"odinger operators associated with substitution dynamical systems in higher dimensions. Focusing on periodic approximations generated by iterating substitutions on initial configurations, we analyze how…
We consider the Schr\"odinger operator with a periodic potential on quasi-1D models of armchair single-wall nanotubes. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite…
We consider kernel operators defined by a dynamical system. The Hausdorff distance of spectra is estimated by the Hausdorff distance of subsystems. We prove that the spectrum map is $ \frac{1}{2} $-H\"older continuous provided the group…
We prove new criteria of stability of the absolutely continuous spectrum of one-dimensional Schr\"odinger operators under slowly decaying perturbations. As applications, we show that the absolutely continuous spectrum of the free and…
We investigate the spectral properties of Schr\"odinger operators in l^2(Z) with limit-periodic potentials. The perspective we take was recently proposed by Avila and is based on regarding such potentials as generated by continuous sampling…
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…
All stationary solutions to the one-dimensional nonlinear Schroedinger equation under box and periodic boundary conditions are presented in analytic form. We consider the case of repulsive nonlinearity; in a companion paper we treat the…
This paper is concerned with the discrete spectrum of the self-adjoint realization of the semi-classical Schr\"odinger operator with constant magnetic field and associated with the de Gennes (Fourier/Robin) boundary condition. We derive an…
Nonlinear Schr\"odinger equation, complemented by a confining potential, possesses a discrete set of stationary solutions. These are called coherent modes, since the nonlinear Schr\"odinger equation describes coherent states. Such modes are…