V. N. Shapovalov

Separation of variables in the stationary Schrödinger equation

The problem of separation of variables in the stationary Schrödinger equation is considered for a charge moving in an external electromagnetic field. On the basis of the definition formulated, necessary and sufficient conditions are found for separation of variables in equations of elliptic type to which the stationary Schrödinger equation belongs. Application of general theorems made it possible to enumerate all types of electromagnetic fields and systems of coordinates in which separation of variables in the stationary Schrödinger equation is possible. Systems of ordinary differential equations which the wave function in the separated variables satisfies are written down to explicit form.

Electron motion in longitudinal electromagnetic fields

A complete study of a charged particles motion in such electromagnetic fields, where vectors of electric and magnetic fields are parallel to each other and to some constant vector, is made. It lists all classes of fields, permitting the solution of the problem by the method of separation of variables, and all concrete kinds of fields in which Dirac’s and Klein–Gordon’s equations can be expressed through the known specifications and are given for the first time.

New exact solutions of the Dirac equation

The search for new exact solutions to the Dirac and Klein-Gordon equations initiated in [1] is continued. New solutions are found for axisymmetric fields and one type of nonstationary field of special configuration. The basic notation and system of units of [1] are retained.

New exact solutions of the Dirac equations. VI

This paper is a continuation of the studies begun in [1–8] to find new exact solutions of the classical relativistic equations of motion and the Klein-Gordon and Dirac equations for a charge moving in an external electromagnetic field.

New exact solutions of the Dirac equation. IV

This paper continues a series of articles [1–3] on new exact solutions of the Dirac equation; it contains solutions of the equation for four cases of crossed fields constituting combinations of the field of a plane, linearly polarized wave and a certain field depending on y and x0−z in a complicated manner.

New exact solutions of the Dirac equation. V

Four types of external electromagnetic fields allowing an exact solution of the Dirac and Klein — Gordon equations are considered. The motion of the electron in such fields is of a specific character; however, from the mathematical point of view some of the problems reduce to cases already studied in earlier papers [1–4].

Symmetry of the Maxwell equations

An analysis is made of the properties of the ring allowed by the Maxwell system, whose “important” generators are first-order differential operators.

New exact solutions of the dirac equation. XII

The search for exact solutions of the Dirac equation begun in [1] is continued. We find three new types of external electromagnetic fields where the Dirac equation, Klein-Gordon equation, and classical Lorentz equation can be solved exactly. We find fields for which explicit solutions to the Klein-Gordon equation can be found but for which explicit solutions of the Dirac equation cannot.

New exact solutions to Dirac's equation. Part 8

This paper continues the research on exact solutions to the Dirac, Klein-Gordon, and Lorentz equations for a charge in an external electromagnetic field [1], The symbols of that paper are used. The fields discussed here do not allow the variables to be separated in Diracs equation in the sense of the definition of [2], but solutions to Diracs equation are obtained.

New exact solutions of the Dirac equation. VII

This paper is a continuation of the studies begun in [1] to find exact solutions of the Dirac, Klein-Gordon, and Lorentz equations for an electron moving in new configurations of an external electromagnetic field.