Related papers: Moebius Schrodinger
In this paper we consider the Schr\"odinger equation with power-like nonlinearity and confining potential or without potential. This equation is known to be well-posed with data in a Sobolev space $\H^{s}$ if $s$ is large enough and…
The dynamic reflection probability and the spectral reflection probability for a one-dimensional Schroedinger operator $H = - \Delta + V$ are characterized in terms of the scattering theory of the pair $(H, H_\infty)$ where $H_\infty$ is…
We provide a simple and short proof of a multidimensional Borg-Levinson type theorem. Precisely, we prove that the spectral boundary data determine uniquely the corresponding potential appearing in the Sch\"odinger operator on an admissible…
Consider a two-dimensional domain shaped like a wire, not necessarily of uniform cross section. Let $V$ denote an electric potential driven by a voltage drop between the conducting surfaces of the wire. We consider the operator ${\mathcal…
We prove that the inverse scattering problem for the Schr\"odinger operator with the separable potential can be reduced to the solving of a certain singular integral equation. We establish the uniqueness of the potential corresponding to…
We establish a variational formula for the exponential decay rate of the Green function of Brownian motion evolving in a random stationary and ergodic nonnegative potential. Such a variational formula is established by Schroeder in 'Green's…
In this short note, we prove positivity of the Lyapunov exponent for 1D continuum Anderson models by leveraging some classical tools from inverse spectral theory. The argument is much simpler than the existing proof due to…
We consider the Schr\"odinger operator $H$ with a periodic potential $p$ plus a compactly supported potential $q$ on the half-line. We prove the following results: 1) a forbidden domain for the resonances is specified, 2) asymptotics of the…
We study a non-linear Schroedinger equation with a Hartree-type nonlinearity and a localized random time-dependent external potential. Sharp dispersive estimates for the linear Schroedinger equation with a random time-dependent potential…
We prove that 3-dimensional Schrodinger operator with slowly decaying potential has an absolutely continuous spectrum that fills the positive half-line. The asymptotics of Green's function is obtained as well.
We consider Schr\"odinger operators on [0,\infty) with compactly supported, possibly complex-valued potentials in L^1([0,\infty)). It is known (at least in the case of a real-valued potential) that the location of eigenvalues and resonances…
This paper extends Remling's Theorem to vector-valued discrete Schrodinger operators, showing that the {\omega} limit points of the matrix potentials, under the shift map, are reflectionless on the absolutely continuous spectrum with full…
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 localisation effects of strong disorder on the spectral and dynamical properties of (matrix and scalar) Schroedinger operators with non-monotone random potentials, on the d-dimensional lattice. Our results include dynamical…
In this article we consider the one-dimensional Schrodinger operator L(Q) with a Hermitian periodic m by m matrix potential Q. We investigate the bands and gaps of the spectrum and prove that the main part of the positive real axis is…
We improve the Lieb-Thirring type inequalities by Demuth, Hansmann and Katriel (J. Funct. Anal. 2009) for Schr\"odinger operators with complex-valued potentials. Our result involves a positive, integrable function. We show that in the…
We study an inverse scattering problem associated with a Schr\"odinger system where both the potential and source terms are random and unknown. The well-posedness of the forward scattering problem is first established in a proper sense. We…
Consider operators $L^{V}:=\Delta + V$ in a bounded Lipschitz domain $\Omega \subset \mathbb{R}^N$. Assume that $V\in C^{1,1}(\Omega)$ and $V$ satisfies $V(x) \leq \overline{a} \mathrm{dist}(x,\partial\Omega)^{-2}$ in $\Omega$ and a second…
The Nonlinear Schroedinger Equation (NLSE) with a random potential is motivated by experiments in optics and in atom optics and is a paradigm for the competition between the randomness and nonlinearity. The analysis of the NLSE with a…
The first author established in [8] a quantitative Borg-Levinson theorem for the Schr\"odinger operator with unbounded potential. In the present work, we extend the results in [8] to the magnetic Schr\"odinger operator. We discuss both the…