Asymptotics of the Spectra of Schrödinger and Dirac Operators with Strong Magnetic Field

Abstract

In this chapter we find eigenvalue asymptotics for two- and three-dimensional Schrödinger and Dirac operators in situations in which the role of the magnetic field is important. We saw in Chapters 6 and 7 that these four operators are essentially different. Moreover, it is convenient to separate the case of constant spectral parameter τ from the case in which τ tends to some specific limit τ* in a given situation. Therefore this chapter is divided into sections in the following way: in sections 11.1 and 11.2, two- and three-dimensional (respectively) Schrödinger and Dirac operators are treated for fixed spectral parameter τ, and in sections 11.3–11.5, the same operators are treated for τ tending to a specific limit; the additional section appears because two-dimensional Schrödinger and Dirac operators are treated separately for the variable τ in section 11.3 and section 11.4 respectively.