Constructive Determination of the Spectral Invariants

In this chapter describes the constructive determinationConstructive determination of the spectral invariantsSpectral invariants explicitly expressed with respect to the Fourier coefficientsFourier coefficients of the potential by using the Bloch eigenvalues as input data. At the same time, it gives a rich set of invariants that is enough to determine the potential q. This chapter consists of 5 sections. First section is the introduction and preliminary facts where we discuss the related papers, describe briefly the scheme of this chapter and recall the results of Chap. 3 which are used essentially in this chapter. In Sect. 4.2, we develop the asymptotic formulas obtained in Chap. 3 and write the first and second term of the asymptotic formulasAsymptotic formulas for the the Bloch eigenvalues in the explicit form. In Sect. 4.3, we investigate the derivatives of the band functions \(\Lambda _{n}\) with respect to the quasimomentum. In Sect. 4.4, using the results of the previous sections, we determine constructively a family of spectral invariants of this operator from the given Bloch eigenvalues. Some of these invariants generalize the well-known invariants and others are entirely new. The new invariants are explicitly expressed by Fourier coefficientsFourier coefficients of the potential which present the possibility of determining the potential constructively by using the Bloch eigenvalues as input data in the next chapter. Section 4.5 is the Appendix, where we give some estimations and calculations of previous sections.