Zero radius interaction for the biharmonic and polyharmonic equations
type: Item
instance of: scholarly article
title: Zero radius interaction for the biharmonic and polyharmonic equations
zbMATH Open document ID: 0641.35022
DOI: 10.1007/BF01159567
author: Boris S. Pavlov
author: Yulia Karpeshina
published in: Mathematical Notes
publication date: 1986-01-01T00:00:00Z
review text: In the paper by \textit{F. A. Berezin} and \textit{L. D. Faddeev} [Sov. Math., Dokl. 2, 372-375 (1961); translation from Dokl. Akad. Nauk SSSR 137, 1011-1014 (1961; Zbl 0117.066)] there is presented a strict version of zero radius potentials for the Schrödinger equation in which it is shown that such potentials can be described correctly within the framework of the theory of extensions. As is well known, zero radius potentials make it possible to model many interesting problems for the Schrödinger equation and, sometimes, to even carry out concrete physical calculations [see, for example, \textit{Yu. N. Demkov} and \textit{V. N. Ostrovskij}, Method of zero radius potentials in atomic physics, Leningrad State Univ. (1975)]. We introduce the concept of zero radius interaction for the biharmonic equation and, later, for the polyharmonic equation. We have here, in fact, not a single parameter real family of extensions, as in the case of the Schrödinger equation, but a family depending on several parameters. With the help of such interactions we can, in the biharmonic case, for example, describe the oscillations of a two-dimensional elastic continuum with various point defects. In point of fact, the theory of extensions enables us to give a complete classification of point defects and to write out in explicit form the solutions of the corresponding planar problems of the theory of elasticity. The various real point defects in this case can be interpreted as different forms of dislocations whose axes are orthogonal to a deformable plane.
Mathematics Subject Classification ID: 35J40
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 4044321
zbMATH Keywords: zero radius potentials
zbMATH Keywords: Schrödinger equation
zbMATH Keywords: zero radius interaction
zbMATH Keywords: biharmonic equation
zbMATH Keywords: polyharmonic equation
zbMATH Keywords: several parameters
zbMATH Keywords: elasticity
MaRDI profile type: Publication
cites work: Q5726865
cites work: Q4157100
cites work: Q3278549
description: scientific article; zbMATH DE number 4044321