Zero radius interaction for the biharmonic and polyharmonic equations
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.
scientific article; zbMATH DE number 4044321