On Absolute Continuity of the Periodic Schrödinger Operators

Abstract

for some basis {ej}j=1 of R . We are interested in the spectral properties of the periodic Schrodinger operator −∆+ V(x) in R. When d = 3, L. Thomas [20] proved that the spectrum of −∆ + V is purely absolutely continuous if V ∈ Lloc(R). Thomas’s approach plays an important role in the subsequent development. In the book by M. Reed and B. Simon [15], it was used to show that −∆+ V is absolutely continuous if V ∈ Lloc(R), where r ≥ d− 1 if d ≥ 4 and where r = 2 if d = 2 or 3. In [4] the basic idea ofThomas was applied to the Dirac operator with periodic potential. Recently, the absolute continuity of the magnetic Schrodinger operator (−i∇−a(x))2 +V(x),with periodic potentials a and V,was investigated by R. Hempel and I. Herbst [6] , [7] , M. Birman and T. Suslina [1] , [2] , A. Morame [13], and A. Sobolev [16]. In particular, the results in [2] , pertaining to the case a = 0, give the absolute continuity of −∆+ V when d = 2 and V ∈ Lloc(R) for some r > 1. Very recently, Birman and Suslina [3] established the absolute continuity of −∆+V for d = 3, V ∈ L loc (R), and for d ≥ 4, V ∈ L loc (R). In this paperweprove that the spectrumof−∆+V is purely absolutely continuous if d ≥ 3 and V ∈ L loc (R). This improves the results in [3] when d ≥ 5. In the context of