相关论文: On separable Schr\"odinger equations

The paper studies existence of solutions for the nonlinear Schr\"odinger equation with a general bounded external magnetic field. In particular, no lattice periodicity of the magnetic field or presence of external electric field is…

We prove that the Schr\"odinger equation for N number of particles in the time dependent electro-magnetic field generates a unique unitary propagator on the state space under the condition that the field is smooth and moderately but almost…

In this paper, we establish a Paley-Wiener type uncertainty principle for Schr\"odinger equations with bounded electric and magnetic potentials, \begin{align*} i\partial_tu+\Delta_Au+V(t,x)u=0,\,\,u(0,x)=u_0(x), \end{align*} where…

We consider the three-dimensional Dirac equation in spherical coordinates with coupling to static electromagnetic potential. The space components of the potential have angular (non-central) dependence such that the Dirac equation is…

We derive the Schroedinger equation for a spinless charged particle constrained to a curved surface with electric and magnetics fields applied. The particle is confined on the surface using a thin-layer procedure, giving rise to the…

Found all equivalence classes for electromagnetic potentials and space-time metrics of Stackel spaces, provided that the equations of motion of the classical charged test particles are integrated by the method of complete separation of…

In a previous paper$^1$, submitted to Journal of Physics A -- we presented an infinite class of potentials for which the radial Schr\"odinger equation at zero energy can be solved explicitely. For part of them, the angular momentum must be…

Carter derived the forms of the metric and the vector potentials of the space-times in which the relativistic Schrodinger equation for the motion of a charged particle separates. Here we show that on each `spheroidal' surface a rotation…

The necessary and sufficient conditions for a function to be totally or partially separable are derived. It is shown that a function is totally separable if and only if each component of the gradient vector of depends only on the…

The aim of the article to clarify the status of Shapiro plane wave solutions of the Schr\"odinger's equation in the frames of the well-known general method of separation of variables. To solve this task, we use the well-known cylindrical…

We present a systematic approach for the separation of variables for the two-dimensional Dirac equation in polar coordinates. The three vector potential, which couple to the Dirac spinor via minimal coupling, along with the scalar potential…

We relax the usual diagonal constraint on the matrix representation of the eigenvalue wave equation by allowing it to be tridiagonal. This results in a larger solution space that incorporates an exact analytic solution for the non-central…

We analyze one particle, two-center quantum problems which admit separation of variables in prolate spheroidal coordinates, a natural restriction satisfied by the H$_2^+$ molecular ion. The symmetry operator is constructed explicitly. We…

We obtain analytic solution of the time-independent Schrodinger equation in two dimensions for a charged particle moving in the field of an electric quadrupole. The solution is written as a series in terms of special functions that support…

We consider the problem of separation of variables in the Kramers equation admitting a non-trivial symmetry group. Provided the external potential $V(x)$ is at most quadratic, a complete solution of the problem of separation of variables is…

We study the valence electron of an alkaline atom or a general charged particle with arbitrary spin and with magnetic moment moving in a rotating magnetic field. By using a time-dependent unitary transformation, the Schr\"odinger equation…

Schr\"odinger equations with nonlinearities concentrated in some regions of space are good models of various physical situations and have interesting mathematical properties. We show that in the semiclassical limit it is possible to…

The Maxwell equations for the spherical components of the electromagnetic fields outside sources do not separate into equations for each component alone. We show, however, that general solutions can be obtained by separation of variables in…

This article is devoted to the spectral analysis of the electro-magnetic Schr\"odinger operator on the Euclidean plane. In the semiclassical limit, we derive a pseudo-differential effective operator that allows us to describe the spectrum…

Solvable Natanzon potentials in nonrelativistic quantum mechanics are known to group into two disjoint classes depending on whether the Schr\"odinger equation can be reduced to a hypergeometric or a confluent hypergeometric equation. All…