Asymptotic Formulas for the Bloch Eigenvalues and Bloch Functions

In this chapter, we construct a perturbation theory for the multidimensional Schrödinger operatorSchrödinger operator with a periodic potential. This chapter consists of six sections. The first section is the introduction, where we define the non-resonance and resonance domainsResonance domains U and V,  describe briefly the scheme of this chapter and discuss the related papers. The asymptotic formulasAsymptotic formulas of arbitrary order for the Bloch eigenvalues when the corresponding quasimomentum lies in the non-resonance and resonance domainsResonance domains are obtained in Sects. 3.2 and 3.3, respectively. In Sect. 3.4, we obtain asymptotic formulasAsymptotic formulas for the Bloch functions when the quasimomentum lies in a set \(B\subset U\) which has asymptotically full measure in the momentum (reciprocal) space. In Sect. 3.5, we construct and investigate the large part of the isoenergetic surfaces in the high energy regionHigh energy region which implies the validity of the Bethe–Sommerfeld conjecture. Note that the method of this chapter is the first and unique by which the asymptotic formulasAsymptotic formulas for the Bloch eigenvalues and Bloch functions and the validity of the conjecture for arbitrary lattice and arbitrary dimension were proved. In Sect. 3.6, we obtain the asymptotic formulasAsymptotic formulas for the Bloch functions when the corresponding quasimomentum lies in a set \(B_{\delta }\subset V\) which is near to the diffraction hyperplaneDiffraction hyperplane  \(D_{\delta }\) and is constructed so that it can be easily used for the constructive determination (in Chap. 4 ) of a family of the spectral invariantsSpectral invariants by the given Bloch eigenvalues.