Spectral asymptotics for Schrodinger operators with periodic point interactions

Spectrum of the second-order differential operator with periodic point interactions in L-2(R) is investigated. Classes of unitary equivalent operators of this type are described. Spectral asymptotics for the whole family of periodic operators are calculated. It is proven that the first several terms in the asymptotics determine the class of equivalent operators uniquely. It is proven that the spectrum of the operators with anomalous spectral asymptotics (when the ratio between the lengths of the bands and gaps tends to zero at infinity) can be approximated by standard periodic "weighted" operators with step-wise density functions. It is shown that this sequence of periodic weighted operators converges in the norm resolvent sense to the formal (generalized) resolvent of the periodic "Schrodinger operator" with certain energy-dependent boundary conditions. ne operator acting in an extended Hilbert space such that its resolvent restricted to L-2(R) coincides with the formal resolvent is constructed explicitly. (C) 2002 Elsevier Science.