Related papers: Uniqueness for discrete Schrodinger evolutions
In this paper we give a discrete version of Hardy's uncertainty principle, by using complex variable arguments, as in the classical proof of Hardy's principle. Moreover, we give an interpretation of this principle in terms of decaying…
In this work we shall review some of our recent results concerning unique continuation properties of solutions of Schr\"odinger equations. In this equations we include linear ones with a time depending potential and semi-linear ones.
We consider d-dimensional time dependent Schr\"odinger equations on the Hilbert space of square integrable functions. We assume magnetic and scalar potentials are almost critically singular with respect to spatial variables both locally and…
We investigate the decay estimates of global solutions for a class of one-dimensional inhomogeneous nonlinear Schr\"odinger equations. While most existing results focus on spatial dimensions $d\geq2$, the decay properties in one dimension…
We prove that in 1-D the growth of Sobolev norms for time-dependent linear Schr\"odinger equations is at most logarithmic in time for any (fixed) potential which is analytic (or Gevrey). Recently it was proven in [N] that almost surely the…
We analyse the exact solutions of a conditionally-solvable Schr\"odinger equation with a rational potential. From the nodes of the exact eigenfunctions we derive a connection between the otherwise isolated exact eigenvalues and the actual…
An algorithm for the numerical solution of the Schr\"odinger equation in the case of a time dependent potential is proposed. Our simple modification upgrades the well known method of Koonin while negligibly increasing the computing time. In…
In this short note, we prove Strichartz estimates for Schr\"odinger operators with slowly decaying singular potentials in dimension two. This is a generalization of the recent results by Mizutani, which are stated for dimension greater than…
We prove Strichartz-type estimates for Schroedinger's equation with time-dependent potentials. The time derivative of the potentials need not be integrable, so the total variation of the potentials may be infinite.
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 consider the defocusing nonlinear Schr{\"o}dinger equation in several space dimensions, in the presence of an external potential depending on only one space vari-able. This potential is bounded from below, and may grow arbitrarily fast…
In this paper we obtain a stabilization result for the Schr\"odinger equation under generic assumptions on the potential. Then we consider the Schr\"odinger equation with a potential which has a random time-dependent amplitude. We show that…
The discrete one-dimensional Schr\"odinger operator is studied in the finite interval of length $N=2 M$ with the Dirichlet boundary conditions and an arbitrary potential even with respect to the spacial reflections. It is shown, that the…
We consider the cubic nonlinear Schr\"odinger equation with an exceptional potential. We obtain a sharp time decay for the global in time solution and we get the large time asymptotic profile of small solutions. We prove the existence of…
We consider the discrete versions of the well known Borg theorem and use simple linear algebraic techniques to obtain new versions of the discrete Borg type theorems. To be precise, we prove that the periodic potential of a discrete…
We find exact solutions of the time-dependent Schr\"odinger equation for a family of quasi-exactly solvable time-dependent potentials by means of non-unitary gauge transformations.
Analytical solutions to the time-dependent Schrodinger equation describing a driven two-level system are invaluable to many areas of physics, but they are also extremely rare. Here, we present a simple algorithm that generates an unlimited…
Asymptotics of solutions to relativistic fractional elliptic equations with Hardy type potentials is established in this paper. As a consequence, unique continuation properties are obtained.
For general second order evolution equations, we prove an optimal condition on the degree of unboundedness of the damping, that rules out finite-time extinction. We show that control estimates give energy decay rates that explicitly depend…
We consider the large time behavior of solutions to defocusing nonlinear Schrodinger equation in the presence of a time dependent external potential. The main assumption on the potential is that it grows at most quadratically in space,…