Related papers: Variable potentials and dispersive equations
We show that for a one-dimensional Schr\"odinger operator with a potential whose (j+1)'th moment is integrable the j'th derivative of the scattering matrix is in the Wiener algebra of functions with integrable Fourier transforms. We use…
In this paper, we focus on a general class of Schr\"odinger equations that are time-dependent and quadratic in X and P. We transform Schr\"odinger equations in this class, via a class of time-dependent mass equations, to a class of solvable…
We analyze dynamical properties of the logarithmic Schr{\"o}dinger equation under a quadratic potential. The sign of the nonlinearity is such that it is known that in the absence of external potential, every solution is dispersive, with a…
A derivation of the time-dependent Schr\"odinger equation from the time-independent one is considered. Instead of time, the coordinate of an additional degree of freedom, the clock, is introduced into the original time-independent…
We study solutions to the Cauchy problem for the linear and nonlinear Schroedinger equation with a quadratic Hamiltonian depending on time. For the linear case the evolution operator can be expressed as an integral operator with the…
We prove some multiplicity results for a nonlinear equation of Schroedinger type with potential functions
We analyze the solutions of the Schr\"odinger equation with the low frequency initial data and a time-dependent weakly random potential. We prove a homogenization result for the low frequency component of the wave field. We also show that…
In this paper, we explore the relations between different kinds of Strichartz estimates and give new estimates in Euclidean space $\mathbb{R}^n$. In particular, we prove the generalized and weighted Strichartz estimates for a large class of…
We prove a sharp, global-in-time Strichartz estimate for the Schr\"odinger equation on the cylinder $\mathbb{R}\times\mathbb{T}$.
It is studied time dependence of the evolution operator kernel for the Schr\"odinger equation with a help of the Schwinger -- DeWitt expansion. For many of potentials this expansion is divergent. But there were established nontrivial…
We discuss the observability of a one-dimensional Schr\"odinger equation on certain time dependent domain. In linear moving case, we give the exact boundary and pointwise internal observability for arbitrary time. For the general moving, we…
Fundamental solution for a Schr\"odinger equation with a time-dependent potential of long-range type is constructed. The solution is given as a Fourier integral operator with a symbol uniformly bounded global in time, when measured in…
We present a time dependent quantum perturbation result, uniform in the Planck constant, for perturbations of potentials whose gradients are Lipschitz continuous by potentials whose gradients are only bounded a.e.. Though this low…
We study the hyperboloidal initial value problem for the one-dimensional wave equation perturbed by a smooth potential. We show that the evolution decomposes into a finite-dimensional spectral part and an infinite-dimensional radiation…
In this paper, we determine the C-infinity type wave front sets of the solutions to the Schrodinger equations with time-dependent variable coefficients and potentials by using the wave packet transform. We introduce the infinite sum and…
We prove some local smoothing estimates for the Schr\"{o}dinger initial value problem with data in $L^2(\mathbb{R}^d)$, $d \geq 2$ and a general class of potentials. In the repulsive setting we have to assume just a power like decay…
We obtain an exact solution of the time-dependent Schroedinger equation for a two-electron system confined to a plane by an isotropic parabolic potential whose curvature is periodically modulated in time. From this solution we compute the…
In this article we prove a reducibility result for the linear Schr\"odinger equation on a Zoll manifold with quasi-periodic in time pseudo-differential perturbation of order less or equal than $1/2$. As far as we know, this is the first…
We prove global Strichartz estimates (with spectral cutoff on the low frequencies) for non trapping metric perturbations of the Schroedinger equation, posed on the Euclidean space.
A simple and explicit technique for the numerical solution of the two-particle, time-dependent Schr\"{o}dinger equation is assembled and tested. The technique can handle interparticle potentials that are arbitrary functions of the…