theory of Group Representations and its applications to Quantum Mechanics; Spectral theory for the linear differential operators; N-variable analogues of the classical orthogonal polynomials.
For two-dimensional Schrodinger operator H in periodic magnetic field $B(x,y)$ and electric field with periodic potential $V(x,y)$ are studied typical dispersion laws $\lambda_j(p_1, p_2)$ and their topological characteristics (quantum numbers) are established. The following theorem is proved: in general case, the Schrodinger operator possesses a countable set of dispersion laws with arbitrary quantum numbers which are no way related to each other or to the flux of the external magnetic field.
Graduated from Faculty of Mathematics and Mechanics of M. V. Lomonosov Moscow State University (MSU) in 1978; the post graduated course of Department of Mathematics from Faculty of Mathematics and Mechanics of M. V. Lomonosov Moscow State University in 1981 (department of differential geometry and topology) Ph.D. thesis was defended in 1981.
Lyskova A. S. Topologicheskie kharakteristiki operatora Shredingera v magnitnom pole i slabom potentsiale. // Teor. i matem. fizika, 1985, t. 65, # 3.
Lyskova A. S. Ortogonalnye mnogochleny neskolkikh peremennykh // DAN SSSR, 1991, t. 316, # 6, 1301–1306.
A. S. Lyskova, “On some properties of orthogonal polynomials in several variables”, Uspekhi Mat. Nauk, 52:4(316) (1997), 207–208; Russian Math. Surveys, 52:4 (1997), 840–841
A. S. Lyskova, “Polynomial solutions of a certain class of ordinary differential equations”, Uspekhi Mat. Nauk, 51:3(309) (1996), 207–208; Russian Math. Surveys, 51:3 (1996), 557–558
A. S. Lyskova, “Topological characteristics of the spectrum of the Schrödinger operator in a magnetic field and in a weak potential”, TMF, 65:3 (1985), 368–378; Theoret. and Math. Phys., 65:3 (1985), 1218–1225
A. S. Lyskova, “Topological properties of the Schrödinger operator in a magnetic field and with a weak potential”, Uspekhi Mat. Nauk, 36:5(221) (1981), 181–182; Russian Math. Surveys, 36:5 (1981), 165–166
A. S. Lyskova, “On the Schrödinger operator in a magnetic field”, Uspekhi Mat. Nauk, 36:2(218) (1981), 189–190; Russian Math. Surveys, 36:2 (1981), 181–182