Related papers: Dispersive Estimates for higher dimensional Schr\"…

Let $H:=-\Delta+V$ be a nonnegative Schr\"odinger operator on $L^2({\bf R}^N)$, where $N\ge 2$ and $V$ is a radially symmetric inverse square potential. Let $\|\nabla^\alpha e^{-tH}\|_{(L^{p,\sigma}\to L^{q,\theta})}$ be the operator norm…

In the present work we consider in $L^2(\mathbb{R}_+)$ the Schr\"odinger operator $\mathrm{H_{X,\alpha}}=-\mathrm{\frac{d^2}{dx^2}}+\sum_{n=1}^{\infty}\alpha_n\delta(x-x_n)$. We investigate and complete the conditions of self-adjointness…

We study the distribution of eigenvalues of the one-dimensional Schr\"odinger operator with a complex valued potential $V$. We prove that if $|V|$ decays faster than the Coulomb potential, then the series of imaginary parts of square roots…

We consider a Schr\"odinger hamiltonian $H(A,a)$ with scaling critical and time independent external electromagnetic potential, and assume that the angular operator $L$ associated to $H$ is positive definite. We prove the following: if…

Let $V$ be a {\em periodic} potential on $\RR^3$ that is smooth everywhere except at a discrete set $\maS$ of points, where it has singularities of the form $Z/\rho^2$, with $\rho(x) = |x - p|$ for $x$ close to $p$ and $Z$ is continuous,…

We provide examples of operators $T(D)+V$ with decaying potentials that have embedded eigenvalues. The decay of the potential depends on the curvature of the Fermi surfaces of constant kinetic energy $T$. We make the connection to…

We study the spectra of non-selfadjoint first-order operators on the interval with non-local point interactions, formally given by ${i\partial_x+V+k\langle \delta,\cdot\rangle}$. We give precise estimates on the location of the eigenvalues…

In this paper we study the linear and nonlinear Schr\"odinger equations associated with the Ornstein-Uhlenbeck (OU) operator endowed with the Gaussian measure. While classical Strichartz estimates are well-developed for the free…

In this paper we consider the 1D Schr\"odinger operator $H$ with periodic point interactions. We show an $L^1-L^\infty$ bound for the time evolution operator $e^{-itH}$ restricted to each energy band with decay order $O(t^{-1/3})$ as $t\to…

We study the initial value problem for Schr\"odinger-type equations with initial data presenting a certain Gevrey regularity and an exponential behavior at infinity. We assume the lower order terms of the Schr\"odinger operator depending on…

In this paper we study on $L^2(\mathbb{R}^d)$ the quasi-periodic Schr\"odinger operator $H=-\Delta+ \lambda V(x),$ where $V$ is a real analytic quasi-periodic function and $\lambda>0$. We first show that $H$ has no eigenvalues in…

We consider a family of multi-dimensional Schr\"odinger operators $-\Delta+t V$ with a real $t$. The potential $V$ in our model decays at infinity in a special way, so that it satisfies a certain integral condition. We prove that the…

In this article we study the semi-classical distribution of complex zeros of the eigenfunctions of the 1D Schr\"odinger operators for the class of polynomial potentials of even degree, when an energy level E is fixed.

Let H be a Schr\"odinger operator defined on an unbounded domain D in R^d with Dirichlet boundary conditions (D may equal R^d in particular). Let u(x,E) be a solution of the Schr\"odinger equation (H-E)u(x,E)=0, and let B_R denote a ball of…

Assuming the negative part of the potential is uniformly locally $L^1$, we prove a pointwise $L^p$ estimate on derivatives of eigenfunctions of one-dimensional Schrodinger operators. In particular, if an eigenfunction is in $L^p$, then so…

In this paper, we investigate negative eigenvalues of exactly solvable quantum models, particularly one-dimensional Hamiltonians with $\delta'$-like potentials used to represent localized dipoles. These operators arise as norm resolvent…

In this paper, we investigate dispersive estimates for the time evolution of Hamiltonians $$ H=-\Delta+\sum_{j=1}^N\langle\cdot\,, \varphi_j\rangle \varphi_j\quad\,\,\,\text{in}\,\,\,\mathbb{R}^d,\,\, d\ge 1, $$ where each $\varphi_j$…

We prove a probabilistic level-spacing estimate at the bottom of the spectrum for continuum alloy-type random Schr\"odinger operators, assuming sign-definiteness of a single-site bump function and absolutely continuous randomness. More…

The purpose of this paper is to study spectral properties of non-self-adjoint Schr\"odinger operators $-\Delta-\frac{(n-2)^2}{4|x|^{2}}+V$ on $\mathbb{R}^n$ with complex-valued potentials $V\in L^{p,\infty}$, $p>n/2$. We prove Keller type…

We consider non-local Schr\"odinger operators with kinetic terms given by several different types of functions of the Laplacian and potentials decaying to zero at infinity, and derive conditions ruling embedded eigenvalues out. Our goal in…