Related papers: Analysis of Schr\"odinger means
The method of solving of nonlinear Schr\"odinger equation is considered. Some examples of its applications are demonstrated.
We show that the resonance counting function for a Schr\"odinger operator has maximal order of growth for generic sets of real-valued, or complex-valued, $L^\infty$-compactly supported potentials.
For $f \in \mathscr{S}^2(\mathcal S)_{o}$, the collection of radial $L^2$-Schwartz class functions on Damek--Ricci spaces $\mathcal S$, we consider the Schr\"odinger maximal function, \begin{equation*} S^* f(x):=…
The fundamental solution of the Schr\"odinger equation for a free particle is a distribution. This distribution can be approximated by a sequence of smooth functions. It is defined for each one of these functions, a complex measure on the…
The integrating factor technique is widely used to solve numerically (in particular) the Schr\"odinger equation in the context of spectral methods. Here, we present an improvement of this method exploiting the freedom provided by the gauge…
In this paper, we combine the argument of [12] and [27] to prove the maximal estimates for fractional Schr\"odinger equations $(i\partial_t+\mathcal{L}_{\mathbf{A}}^{\frac\alpha 2})u=0$ in the purely magnetic fields which includes the…
In this paper we obtain minimal support properties of solutions of Schr\"odinger equations. We improve previously known conditions on the potential for which the measure of the support of solutions cannot be too small. We also use these…
We obtain sharp estimates for the quasi norm of the maximal function of f when it satisfies certain conditions.
We review some results of calculations, having the property of maximal transcendentality.
We introduce the notion of Schr\"odinger integral operators and prove sharp local and global regularity results for these (including propagators for the quantum mechanical harmonic oscillator). Furthermore we introduce general classes of…
We consider Schr\"odinger operators of the form $H_R = - d^2/ d x^2 + q + i \gamma \chi_{[0,R]}$ for large $R>0$, where $q \in L^1(0,\infty)$ and $\gamma > 0$. Bounds for the maximum magnitude of an eigenvalue and for the number of…
We review the results having the property of maximal transcendentality.
We consider the maximal function of oscillatory integrals and prove a global estimate for radial test functions which is almost sharp with respect to the Sobolev regularity.
We study arbitrary order symmetry operators for the linear Schr\"odinger equations with arbitrary number of spatial variables. We deduce determining equations for coefficient functions of such operators and consider in detail some cases…
We investigate the Schr\"odinger representations of certain infinite-dimensional Heisenberg groups, using their corresponding Wigner transforms.
We present some recent results on the existence of solutions of the Schr\"odinger flows, and pose some problems for further research.
The objective of this paper is to report on recent progress on Strichartz estimates for the Schr\"odinger equation and to present the state-of-the-art. These estimates have been obtained in Lebesgue spaces, Sobolev spaces and, recently, in…
In this article we investigate the nature of the functions, including important double power terms which arise naturally in considering typical nonlinear Schroedinger equations.
The independent solutions of the one-dimensional Schr\"odinger equation are approximated by means of the explicit summation of the leading constituent WKB series. The continuous matching of the particular solutions gives the uniformly valid…
In this paper maximal commutators and commutators of maximal functions with functions of bounded mean oscillation are investigated. New pointwise estimates for them are proved.