Dept. Theor. Phys., Institute of Magnetism, National Academy of Sciences of Ukraine,
36-b, prosp. Vernadsky, 03142 Kiev-142, Ukraine
e-mail: bel@im.imag.kiev.ua

Many-dimensional Schroedinger operator with a separable finite-gap potential

Abstract:

A report presents the results of studies of the spectral properties of many-dimensional Schroedinger operator with separable finite-gap potential along with their applications to a number problems of physics of solids.

A separable finite-gap potential is the simplest possible many-dimensional generalization of a one-dimensional finite-gap one. Its spectral properties  are simple which allows to describe effectively the spectrum, eigenvalues, eigenfunctions and also matrix elements of any observables analytically (e.g. by means of a residue method). Application of these results to physics of solids appears to be very successful for explanation and quantitative description of many phenomena and properties  of solids: the electron energy spectra of solids and Fermi surfaces of metals, the scattering and absorption of electromagnetic and other waves by finite-gap solids, the electron-phonon interaction, the Peierls transition and Froehlich conductivity, the classification of the quasi-one-dimensional conductors, the oscillations in solids due to isospectral deformations of finite-gap potentials etc.

This approach has a number of perspective generalizations.

References:

1. E.D. Belokolos, J.C. Eilbeck, V.Z. Enolskii, M. Salerno. Exact energy bands and Fermi surfaces of separable Abelian potentials. J. Phys. A: Math. Gen., (2001) v. 34, p. 943-954.
2. V.G. Baryakhtar, E.D. Belokolos, A.M. Korostyl. Analytical method for calculating Fermi surfaces of High-Temperature Superconductors. Phys. Metals (1993) v. 13, No. 1, p. 1-11.
3. V.G. Baryakhtar, E.D. Belokolos, A.M. Korostyl. Method of Separable finite-band potentials: A new method for calculating  Electron Energy Structure. Phys. Metals (1993) v. 12, No. 8, p. 829-838.