Topological characteristics of the spectrum of the Schrodinger operator in a magnetic field and in a weak potential

Journal Article · Sun Jun 01 04:00:00 EDT 1986 · Theor. Math. Phys.; (United States)

OSTI ID:6990440

Lyskova, A S

This paper studies the two-dimensional Schrodinger operator H in a periodic magnetic field B(x,y) and in an electric field with periodic potential V(x,y). It is assumed that the functions B(x,y) and V(x,y) are periodic with respect to some lattice in R/sup 2/ and that the m agnetic flux through a unit cell is an integral number. The operator H is represented as a direct integral over the two-dimensional torus of the reciprocal lattice of elliptic self-adjoint operators H /sub p1/, /sub p2/ which possess a discrete spectrum lambda /sub j/ (p/sub 1/,p/sub 2/), j = 0,1,2.... On the basis of an exactly integrable case - the Schrodinger operator in a constant magnetic field - perturbation theory is used to investigate the typical dispersion laws lambda /sub j/ (p/sub 1/,p/sub 2/) and establish their topological characteristics (quantum numbers). A theorem is proved: In the general case, the Schrodinger operator has a coutable number of dispersion laws with arbitrary quantum numbers in no way related to one another or to thflux of the external magnetic field.

• Research Organization: All-Union Correspondence Electrotechnical Communications Institute
• OSTI ID: 6990440
• Journal Information: Theor. Math. Phys.; (United States), Journal Name: Theor. Math. Phys.; (United States) Vol. 65:3; ISSN TMPHA
• Country of Publication: United States
• Language: English

Related Subjects

657002* -- Theoretical & Mathematical Physics-- Classical & Quantum Mechanics
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
BLOCH EQUATIONS
DIFFERENTIAL EQUATIONS
DISPERSION RELATIONS
ELECTRIC FIELDS
ENERGY LEVELS
EQUATIONS
FIELD THEORIES
FUNCTIONS
HERMITE POLYNOMIALS
LATTICE FIELD THEORY
MAGNETIC FIELDS
MAGNETIC FLUX
MATHEMATICAL OPERATORS
MATHEMATICS
PARTIAL DIFFERENTIAL EQUATIONS
PERTURBATION THEORY
POLYNOMIALS
QUANTUM FIELD THEORY
QUANTUM OPERATORS
SCHROEDINGER EQUATION
TOPOLOGY
TWO-DIMENSIONAL CALCULATIONS
WAVE EQUATIONS