Spectral threshold effect: examples

Question

I know that the effect of homogenization can be treated as a spectral threshold effect. I want to know more examples of spectral threshold effects in mathematical physics.

Answer 1

An overview of threshold effects on a band structure is given by Birman and Suslina, in their paper on Second order periodic differential operators (2004). Generically, this refers to cases where it suffices to know the approximate spectral expansion of the initial differential operator near the lower edge $\lambda=0$ of the spectrum. For example, the appearance of a negative discrete spectrum when the operator is perturbed is a threshold effect. An important physical application is the construction of an "effective mass", which describes the curvature of the band structure near the edge.