Non-linear Spectral Problems

Abstract

This Chapter deals with a spectral geometry of non-linear spectral problems (NLSP). Known also as polynomial operator pencils, such problems appear in many physical applications and, in particular, in equations which determine spectra of single-particle modes. A NLSP is defined here as a linear combination of different powers of an eigenvalue multiplied by operator coefficients which do not commute to each other in general. After describing a method how spectra of NLSP can be found one defines a relevant spectral function, a ‘pseudo-trace’. For a rather wide a class of physically motivated NLSP the pseudo-trace has an asymptotic expansion of the same structure as standard heat kernel asymptotics. Much space of this Chapter is devoted to calculations of the expansion coefficients for NLSP which are expressed as local invariant functionals of background fields.