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JOURNALS // Vestnik Tomskogo Gosudarstvennogo Universiteta. Matematika i Mekhanika // Archive

Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2020 Number 67, Pages 5–17 (Mi vtgu797)

MATHEMATICS

Analytical solution of the Schrödinger integral equation

M. A. Bubenchikova, A. M. Bubenchikova, S. Jambaabc, A. V. Lun-Fud, A. S. Chelnokovaa

a Tomsk State University, Tomsk, Russian Federation
b Mongolian University of Science and Technology, Ulaanbaatar, Mongolia
c National University of Mongolia, Ulaanbaatar, Mongolia
d Gazprom Transgaz Tomsk Ltd

Abstract: In this paper, the question about the use of wave dynamics for solving problems of membrane separation of helium isotopes in the gas state at cryogenic temperatures is considered. The dimensionless form of the stationary Schrodinger differential equation is obtained. Following that, the integral representation form of the wave function is written. This form, which is equivalent to the classical equation, is similar to the integral equation with a degenerate core; however, it contains a modulus of the argument with a shift along the real axis. Using the shift operator, the existing exponential function in the Schrödinger integral equation can be split into a differential operator and an exponential function of the argument module which does not contain a shift. The Fourier identity allows reducing the exponent of the modulus of the argument to a regular exponent. Next, based on the general property of a differential operator acting on an exponent, it is possible to calculate the spectral functions of the problem and write down the distribution for the wave function. This distribution is ultimately expressed through the spectra of the potential barrier. Thereafter, the structure and the spectrum of the composite barrier are considered. With the expression determining the reflection coefficient, it is found that the double-barrier system can have a resonant passage of one of the components in the sequence of distances between the layers of the membrane.

Keywords: integral equation, degenerate kernel, shift operator, exponential functions, Fourier identity.

UDC: 517.9

MSC: 45H05

Received: 21.02.2020

DOI: 10.17223/19988621/67/1



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