Related papers: Comment on "New analytic solution of Schr\"odinger…
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…
We find the form of the potential depending on the coordinates and the time such that a solution, $S$, of the Hamilton--Jacobi equation yields an exact solution, $\exp ({\rm i} S/\hbar)$, of the corresponding Schr\"odinger equation.
In the present paper we introduce a new methodology for the construction of numerical methods for the approximate solution of the one-dimensional Schr\"odinger equation. The new methodology is based on the requirement of vanishing the…
A class of Schr\"odinger-type second-order linear differential equations with a large parameter $u$ is considered. Analytic solutions of this type of equations can be described via (divergent) formal series in descending powers of $u$.…
We propose an extension of Wenzel-Kramers-Brillouin (WKB) approximation for solving the Schr\"odinger equation. A set of coupled differential equations is obtained by considering an ansatz of the wave function with an auxiliary condition on…
We prove analyticity of solutions in $\mathbb{R}^{n}$, $n\ge1$, to certain nonlocal linear Schr\"odinger equations with analytic potentials.
We give a more detailed description of the new system of Pl\"ucker-like equations from [4], discuss how it relates to the usual Pl\"ucker equations, and correct a mistake in that article.
We analyze a recent pedagogical proposal for an alternative treatment of the angular part of the Schr\"odinger equation with a central potential. We show that the authors' arguments are unclear, unconvincing and misleading.
We present some recent results on the existence of solutions of the Schr\"odinger flows, and pose some problems for further research.
We propose an exact method for solving a one-dimensional Schr\"odinger equation. An arbitrary potential is represented by the collection of short-width potentials. For building the collection scheme, a new solvable potential is introduced.…
We propose a new approximate series solution of the semiclassical Wigner equation by uniformization of WKB approximations of the Schr\"odinger eigenfunctions.
We prove almost Strichartz estimates found after adding regularity in the spherical coordinates for Schr\"odinger-like equations. The estimates are sharp up to endpoints. The proof relies on estimates involving spherical averages. Sharpness…
This paper is concerned with an efficient numerical method for solving the 1D stationary Schr\"odinger equation in the highly oscillatory regime. Being a hybrid, analytical-numerical approach it does not have to resolve each oscillation, in…
A method to find exact solutions to nonlinear Schr\"odinger equation, defined on a line and on a plane, is found by connecting it with second order linear ordinary differential equation. The connection is essentially made using Riccati…
The recently introduced scheme [20,21] is extended to propose an algebraic non-perturbative approach for the analytical treatment of Schr\"odinger equations with non-solvable potentials involving an exactly solvable potential form together…
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…
We give a survey on recent developments on nonlinear Schr\"odinger equations with dissipative structure based on the authors' recent works.
In [J. Math. Phys. 53, 042105 (2012)], Bay{\i}n claims to prove the consistency of the purported piece-wise solutions to the fractional Schr\"odinger equation for an infinite square well. However, his calculation uses standard contour…
A recently developed expansion method for analytically solving the ground states of strongly coupling Schr\"odinger equations by Friedberg, Lee and Zhao is extended to excited states and applied to the pedagogically important problems of…
The method of solving of nonlinear Schr\"odinger equation is considered. Some examples of its applications are demonstrated.