On the constructive determination of spectral invariants of the periodic Schrodinger operator with smooth potentials

Abstract

In this paper, we develop the asymptotic formulae, obtained in my previous papers, for the band functions and the Bloch functions of the Schrodinger operator with the smooth periodic potentials. Then using these formulae, we determine constructively a family of spectral invariants of this operator from the given band functions. Some of these invariants generalize the well-known invariants and others are entirely new. The new invariants are explicitly expressed by Fourier coefficients of the potential which present the possibility of determining the potential constructively by using the band functions as input data.