Related papers: Schr\"odinger means in higher dimensions
We obtain $L^p$ estimates of the maximal Schr\"odinger operator in $\mathbb R^n$ using polynomial partitioning, bilinear refined Strichartz estimates, and weighted restriction estimates.
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…
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, we establish the almost everywhere convergence of solutions to the Schr\"odinger operator with complex time $ P_{\gamma}f(x,t) $ in higher dimensions, under the assumption that the initial data $f$ belongs to the Sobolev…
The method of solving of nonlinear Schr\"odinger equation is considered. Some examples of its applications are demonstrated.
The set of dynamic symmetries of the scalar free Schr\"odinger equation in d space dimensions gives a realization of the Schr\"odinger algebra that may be extended into a representation of the conformal algebra in d+2 dimensions, which…
In this survey the contemporary results concerning supersymmetries in generalized Schr\"odinger equations are presented. Namely, position dependent mass Sch\"odinger equations are discussed as well as the equations with matrix potentials.…
We present some old and new results on dispersive estimates for Schroedinger equations.
We present some recent results on the existence of solutions of the Schr\"odinger flows, and pose some problems for further research.
We call any measure on a path space, a path measure. Some notions about path measures which appear naturally when solving the Schr\"odinger problem are presented and worked out in detail.
We develop an algebraic approach to studying the spectral properties of the stationary Schr\"odinger equation in one dimension based on its high order conditional symmetries. This approach makes it possible to obtain in explicit form…
The initial value problem is considered for a higher order nonlinear Schr\"odinger equation with quadratic nonlinearity. Results on existence and uniqueness of weak solutions are obtained. In the case of an effective at infinity additional…
We prove dispersive and Strichartz estimates for Schr\"{o}dinger equations on normal real form symmetric spaces. These estimates apply to the well-posedness and scattering for the nonlinear Schr\"{o}dinger equations.
We describe a certain "self-similar" family of solutions to the free Schroedinger equation in all dimensions, and derive some consequences of such solutions for two specific problems.
We obtain new results about the high-energy distribution of resonances for the one-dimensional Schr\"odinger operator. Our primary result is an upper bound on the density of resonances above any logarithmic curve in terms of the singular…
In this paper, we study the almost everywhere convergence results of Schr\"odinger operator with complex time along curves. We also consider the fractional cases. All results are sharp up to the endpoints.
For the first time, a nonlinear Schr\"odinger equation of the general form is considered, depending on time and two spatial variables, the potential and dispersion of which are specified by two arbitrary functions. This equation naturally…
Estimates for the total multiplicity of eigenvalues for Schr\"odinger operator are established in the case of compactly supported or exponentially decreasing complex-valued potential.
A method is presented to construct exactly solvable nonlinear extensions of the Schr\"odinger equation. The method explores a correspondence which can be established under certain conditions between exactly solvable ordinary Schr\"odinger…
Almost everywhere convergence on the solution of Schr\"odinger equation is an important problem raised by Carleson in harmonic analysis. In recent years, this problem was essentially solved by building the sharp $L^p$-estimate of…