Equation de Schrödinger avec champ magnétique et équation de Harper. (The Schrödinger equation with magnetic field and the Harper equation). (French) Zbl 0699.35189

Helffer, B.; Sjöstrand, J.

Schrödinger operators, Proc. Nord. Summer Sch. Math., Sandbjerg Slot, Sønderborg/Denmark 1988, Lect. Notes Phys. 345, 118-197 (1989).

[For the entire collection see Zbl 0686.00012.]
This paper consists of two independent parts. In part I “Approximation of strong magnetic field” a two-dimensional Schrödinger operator \(P_{B,V}\) with a constant intensity B of magnetic field is considered and the investigation of its spectrum is reduced to the investigation of the spectra of the sequence of one-dimensional pseudo-differential operators. For an appropriate potential V, \(2\pi\) periodic and close to \(\cos x_ 1+\cos x_ 2\) and for admissible values B it is proved that Spec\(P_{B,V}\) is a Cantor set of measure 0.
In Part II “Approximation of the weak magnetic field: substitution of Peiers” an n-dimensional Schrödinger operator with a constant tensor intensity of the magnetic field and with periodic potential is considered and the Peiers approximation is justified, density of states and interaction of different spectral gaps are examined. The case when additional symmetries are present also is investigated.

Reviewer: V.I.Ivrij

MSC:

35P05	General topics in linear spectral theory for PDEs
35S05	Pseudodifferential operators as generalizations of partial differential operators
35J10	Schrödinger operator, Schrödinger equation

Keywords:

Harper equation; strong magnetic field; two-dimensional Schrödinger operator; one-dimensional pseudo-differential; Cantor set; weak magnetic field; periodic potential; Peiers approximation; spectral gaps

Citations:

Zbl 0686.00012