Related papers: Dispersive Bounds for the three-dimensional Schrod…

The one-dimensional Schr\"{o}dinger equation for a class of potentials $V(|x|)$ which vanish at infinity and present dominant singularity at the origin in the form $\alpha /|x|^{\beta}$ ($0<\beta \leq 2$) is investigated. The Hermiticity of…

This work is a continuation of our previos paper, where for the Schr\"odinger operator $H=-\Delta+ V(\e)\cdot$ $(V(\e)\ge 0)$, acting in the space $L_2(\R^d)\,(d\ge 3)$, some sufficient conditions for discreteness of its spectrum have been…

We prove an explicit weighted estimate for the semiclassical Schr\"odinger operator $P = - h^2 \partial^2_x + V(x;h)$ on $L^2(\mathbb{R})$, with $V(x;h)$ a finite signed measure, and where $h >0$ is the semiclassical parameter. The proof is…

Using boundary triples, we develop an abstract framework to investigate the complete non-selfadjointness of the maximally dissipative extensions of dissipative operators of the form $S+iV$, where $S$ is symmetric with equal finite defect…

We derive a dispersion estimate for one-dimensional perturbed radial Schr\"odinger operators. We also derive several new estimates for solutions of the underlying differential equation and investigate the behavior of the Jost function near…

In this paper, the scattering and spectral theory of $H=\Delta_g+V$ is developed, where $\Delta_g$ is the Laplacian with respect to a scattering metric $g$ on a compact manifold $X$ with boundary and $V\in C^\infty(X)$ is real; this extends…

We establish global two-sided heat kernel estimates (for full time and space) of the Schr\"odinger operator $-\frac{1}{2}\Delta+V$ on $\R^d$, where the potential $V(x)$ is locally bounded and behaves like $c|x|^{-\alpha}$ near infinity with…

This paper is devoted to the study of the existence of positive and bounded solutions for a Schr\"odinger type equation defined on the entire Euclidean space, involving a general integro-differential operator. We consider the case where the…

Consider the Schrodinger equation -\Delta u =(k+V) u in an infinite slab S= \R^{n-1}x (0,1), where V is a bounded potential supported on a set D of finite measure. We prove necessary conditions for the existence of nontrivial admissible…

Let $L=-\Delta + V(x)$ be a Schr\"odinger operator on $\mathbb R^d$, where $V(x)\geq 0$, $V\in L^2_{\rm loc} (\mathbb R^d)$. We give a short proof of dimension free $L^p(\mathbb R^d)$ estimates, $1<p\leq 2$, for the vector of the Riesz…

Two-particle discrete Schr\"{o}dinger operators $H(k)=H_{0}(k)-V$ on the three-dimensional lattice $\Z^3,$ $k$ being the two-particle quasi-momentum, are considered. An estimate for the number of the eigenvalues lying outside of the band of…

For the two-dimensional Schr\"odinger equation $$ [- \Delta +v(x)]\psi=E\psi,\ x\in \R^2,\ E=E_{fixed}>0 \ \ \ \ \ (*)$$ at a fixed positive energy with a fast decaying at infinity potential $v(x)$ dispersion relations on the scattering…

We study the Schr\"{o}dinger-Poisson-Slater equation \begin{equation*}\left\{\begin{array}{lll} -\Delta u + \lambda u + \big(|x|^{-1} \ast |u|^{2}\big)u = V(x) u^{ p_{\varepsilon}-1 }, \, \text{ in } \mathbb{R}^{3},\\[2mm]…

We prove Strichartz estimates for the Schr\"odinger equation with scaling-critical electromagnetic potentials in dimensions $n\geq3$. The decay assumption on the magnetic potentials is critical, including the case of the Coulomb potential.…

We prove new and explicit formulas for the wave operators of Schroedinger operators in R^3. These formulas put into light the very special role played by the generator of dilations and validate the topological approach of Levinson's theorem…

Let $\Omega$ be a bounded domain in $R^n$ with $C^2$-smooth boundary of co-dimension 1, and let $H=-\Delta +V(x)$ be a Schr\"odinger operator on $\Omega$ with potential V locally bounded. We seek the weakest conditions we can find on the…

Let $\alpha>0$, $H=(-\triangle)^{\alpha}+V(x)$, $V(x)$ belongs to the higher order Kato class $K_{2\alpha}(\mathbbm{R}^n)$. For $1\leq p\leq \infty$, we prove a polynomial upper bound of $\|e^{-itH}(H+M)^{-\beta}\|_{L^p, L^p}$ in terms of…

Let -Delta+V be the Schrodinger operator acting on L^2(R^d,C) with d>2 odd. Here V is a bounded real or complex function vanishing outside the closed ball of center 0 and of radius a. We show for generic potentials V that the number of…

We consider an anisotropic model case for a strictly convex domain of dimension $d\geq 2$ with smoothboundary and we describe dispersion forthe semi-classical Schr{\"o}dinger equation with Dirichlet boundary condition. More specifically, we…

In this note, we prove pointwise decay in time of solutions to the 3D energy-critical nonlinear Schr\"odinger equations assuming data in $L^1\cap H^3$. The main ingredients are the boundness of the Schr\"odinger propagators in Hardy space…