Related papers: Propagation for Schr\"odinger operators with poten…

We introduce new models for Schr\"odinger-type equations, which generalize standard NLS and for which different dispersion occurs depending on the directions. Our purpose is to understand dispersive properties depending on the directions of…

Generalized eigenfunctions of the two-dimensional relativistic Schr\"odinger operator $H=\sqrt{-\Delta}+V(x)$ with $|V(x)|\leq C< x>^{-\sigma}$, $\sigma>3/2$, are considered. We compute the integral kernels of the boundary values…

In this paper we introduce a method that allows one to prove uniform local results for one-dimensional discrete Schr\"odinger operators with Sturmian potentials. We apply this method to the transfer matrices in order to study the Lyapunov…

We prove that the eigenvalues of a continuum random Schr\"odinger operator $-\Delta+ V_{\omega}$ of Anderson type, with complex decaying potential, can be bounded (with high probability) in terms of an $L^q$ norm of the potential for all…

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}…

In this paper, we first investigate the integral curvature condition to extend the mean curvature flow of submanifolds in a Riemannian manifold with codimension $d\geq1$, which generalizes the extension theorem for the mean curvature flow…

In this paper, we consider convergence properties for generalized Schr\"{o}dinger operators along tangential curves in $\mathbb{R}^{n} \times \mathbb{R}$ with less smoothness comparing with Lipschitz condition. Firstly, we obtain sharp…

Consider the one-dimensional discrete Schr\"odinger operator $H_{\theta}$: $$(H_{\theta} q)_n=-(q_{n+1}+q_{n-1})+ V(\theta+n\omega) q_n \ , \quad n\in Z \ ,$$ with $\omega\in R^d$ Diophantine, and $V$ a real-analytic function on $ T^d=(…

We obtain several essential self-adjointness conditions for a Schroedinger type operator D*D+V acting in sections of a vector bundle over a manifold M. Here V is a locally square-integrable bundle map. Our conditions are expressed in terms…

In this paper, we investigate the delocalization property of the discrete Schr\"odinger operator $H_\omega=-\Delta+v_n\omega_n\delta_{n,n'}$, where $v_n=\kappa |n|^{-\alpha}$ and $\omega=\{\omega_n\}_{n\in\mathbb{Z}^d}\in \{\pm…

We prove a unique continuation principle for spectral projections of Schr\" odinger operators. We consider a Schr\" odinger operator $H= -\Delta + V$ on $\mathrm{L}^2(\mathbb{R}^d)$, and let $H_{\Lambda}$ denote its restriction to a finite…

Let $(M,g)$ be a smooth, compact Riemannian manifold and $\{\phi_h\}$ an $L^2$-normalized sequence of Laplace eigenfunctions, $-h^2\Delta_g\phi_h=\phi_h$. Given a smooth submanifold $H \subset M$ of codimension $k\geq 1$, we find conditions…

We study half-line Schr\"odinger operators with locally $H^{-1}$ potentials. In the first part, we focus on a general spectral theoretic framework for such operators, including a Last--Simon-type description of the absolutely continuous…

We prove dispersive estimates for two models~: the adjacency matrix on a discrete regular tree, and the Schr\"odinger equation on a metric regular tree with the same potential on each edge/vertex. The latter model can be thought of as an…

We discuss Schr\"odinger operators on a half-line with decaying oscillatory potentials, such as products of an almost periodic function and a decaying function. We provide sufficient conditions for preservation of absolutely continuous…

We prove continuity results for Fourier integral operators with symbols in modulation spaces, acting between modulation spaces. The phase functions belong to a class of nondegenerate generalized quadratic forms that includes Schr\"odinger…

We prove eigenvalue bounds for Schr\"odinger operator $-\Delta_g+V$ on compact manifolds with complex potentials $V$. The bounds depend only on an $L^q$-norm of the potential, and they are shown to be optimal, in a certain sense, on the…

As a simple model for lattice defects like grain boundaries in solid state physics we consider potentials which are obtained from a periodic potential $V = V(x,y)$ on $\R^2$ with period lattice $\Z^2$ by setting $W_t(x,y) = V(x+t,y)$ for $x…

We investigate $L^1(\mathbb R^n)\to L^\infty(\mathbb R^n)$ dispersive estimates for the Schr\"odinger operator $H=-\Delta+V$ when there is an eigenvalue at zero energy in even dimensions $n\geq 6$. In particular, we show that if there is an…

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 minimum. The main result,…