Weylian Asymptotics of Spectra

Abstract

This chapter is devoted to applications of the results of Chapters 4, 5, 8 and 9 to eigenvalue asymptotics. In section 10.1 we treat semiclassical asymptotics. In sections 10.2–10.4 we consider asymptotics of eigenvalues tending to infinity: in section 10.2 the domain is bounded “in principal” (i.e., exits to infinity are thin enough) and the coefficients are weakly singular; in section 10.3 the coefficients are strongly singular and the coercivity condition is fulfilled but the domain is still bounded “in principal;” in section 10.4 the domain is “very unbounded” and the spectrum is discrete only because the lower order terms are singular. In section 10.5 we treat asymptotics of eigenvalues tending to the boundary of the essential spectrum. In particular, in section 10.4 we treat the Schrödinger operator with potential V tending to +∞ at infinity and with the spectral parameter τ tending to +∞, and in section 10.5 we treat the Schrödinger and Dirac operators with potentials tending to 0 at infinity and the spectral parameter τ tending to −0 and M −0 or − M + 0 respectively (M is the mass in the case of the Dirac operator). Section 10.6 is devoted to multiparametrical asymptotics; for example, there is the semiclassical parameter h → +0 and the spectral parameter τ tending either to +∞ or −∞ (for the Schrödinger and Dirac operators in bounded domains or for the Schrödinger operator in unbounded domains), or to −0, M − 0 or − M + 0 (for the Schrödinger and Dirac operators in unbounded domains), or to inf V + 0, inf V + M + 0, sup V − M − 0 (for the Schrödinger and Dirac operators) where for the Schrödinger operator inf V is not necessarily finite. Moreover, the case of a vector semiclassical parameter h = (h ₁,..., h _d ) is treated.