Related papers: Sharp decay estimates in Lorentz spaces for nonneg…

We generalize and extend results on decay rates of singular values or eigenvalues of positive integral operators from unit spheres to two-point homogeneous spaces. The rates we present depend upon the order of the Laplace-Beltrami operator…

We establish dispersive time-decay estimates for periodic Jacobi operators on the discrete half-line, $\N$. Specifically, we prove $t^{-1/2}$ decay in the weighted $\ell^\infty_{-1}$ norm for all such operators. For the global $\ell^1 \to…

We consider the initial value problem for a system of cubic nonlinear Schr\"odinger equations with different masses in one space dimension. Under a suitable structural condition on the nonlinearity, we will show that the small amplitude…

For a two-dimensional Schr\"odinger operator $H_{\alpha V}=-\Delta-\alpha V,\ V\ge 0,$ we study the behavior of the number $N_-(H_{\alpha V})$ of its negative eigenvalues (bound states), as the coupling parameter $\alpha$ tends to infinity.…

Let $L$ be a non-negative self-adjoint operator acting on $L^2(X)$, where $X$ is a space of homogeneous type with a dimension $n$. Suppose that the heat operator $e^{-tL}$ satisfies the generalized Gaussian $(p_0, p'_0)$-estimates of order…

This paper aims to give a general (possibly compact or noncompact) analog of Strichartz inequalities with loss of derivatives, obtained by Burq, G\'erard, and Tzvetkov [19] and Staffilani and Tataru [51]. Moreover we present a new approach,…

Let $\{e^{-tL^{\alpha}}\}_{t>0}$ be the fractional Schr\"{o}dinger semigroup associated with $L=-\Delta+V$, where $V$ is a non-negatvie potential belonging to the reverse H\"{o}lder class. In this paper, we establish weighted boundedness…

For the Hamiltonian operator H = -{\Delta}+V(x) of the Schr\"odinger Equation with a repulsive potential, the problem of local decay is considered. It is analyzed by a direct method, based on a new, L^2 bounded, propagation observable. The…

This paper investigates the $L^p$-boundedness of wave operators associated with the nonhomogeneous fourth-order Sch\"odinger operator $H = \Delta^2 - \Delta + V(x)$ on $\mathbb{R}^n$. Assuming the real-valued potential $ V $ exhibits…

Let $(\mathbb M, d,\mu)$ be a metric measure space with upper and lower densities: $$ \begin{cases} |||\mu|||_{\beta}:=\sup_{(x,r)\in \mathbb M\times(0,\infty)} \mu(B(x,r))r^{-\beta}<\infty;\\ |||\mu|||_{\beta^{\star}}:=\inf_{(x,r)\in…

We consider three magnetic relativistic Schr\"odinger operators which correspond to the same classical symbol $\sqrt{(\xi-A(x))^2+m^2}+V(x)$ and whose heat semigroups admit the Feynman-Kac-It\^o type path integral representation…

We prove L^1 --> L^\infty estimates for linear Schroedinger equations in dimensions one and three. The potentials are only required to satisfy some mild decay assumptions. No regularity on the potentials is assumed.

Let $\mathcal L=-\Delta_{\mathbb H^n}+V$ be a Schr\"odinger operator on the Heisenberg group $\mathbb H^n$, where $\Delta_{\mathbb H^n}$ is the sub-Laplacian on $\mathbb H^n$ and the nonnegative potential $V$ belongs to the reverse H\"older…

For a bounded real-valued function $V$ on ${\Bbb R}^d$, we consider two Schr\"odinger operators $H_+=-\Delta+V$ and $H_-=-\Delta-V$. We prove that if the negative spectra $H_+$ and $H_-$ are discrete and the negative eigenvalues of $H_+$…

Let $\mathcal{H}=-\Delta_{\mathbb{H}}+V$ be the Schr\"odinger operator on the Heisenberg group $\mathbb{H}^n$, where $\Delta_{\mathbb{H}}$ is the full laplacian on $\mathbb{H}^n$ and $V$ is a positive smooth potential, bounded below and…

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 address the decay rates of the energy for the damped wave equation when the damping coefficient $b$ does not satisfy the Geometric Control Condition (GCC). First, we give a link with the controllability of the associated Schr\"odinger…

The aim of the present paper is twofold:(1) We carry on with developing an abstract method for deriving decay estimates on the semigroup associated to non-symmetric operators in Banach spaces as introduced in [10]. We extend the method so…

We consider eigenvalue sums of Schr\"odinger operators $-\Delta+V$ on $L^2(\R^d)$ with complex radial potentials $V\in L^q(\R^d)$, $q<d$. We prove quantitative bounds on the distribution of the eigenvlaues in terms of the $L^q$ norm of $V$.…

We study the low-energy asymptotics of the spectral shift function for Schr\"odinger operators with potentials decaying like $O(\frac{1}{|x|^2})$. We prove a generalized Levinson's for this class of potentials in presence of zero eigenvalue…