On the Schrödinger operator with periodic point interactions in the three-dimensional case

Abstract

We prove that it is possible to define the self-adjoint operator which gives sense to the merely formal expression −Δ−∑y∈Lλδ(⋅−y) (where L is a certain lattice of R3) as the limit when ε→0+ in the resolvent sense of the net Hε =−Δ+∑y∈L λ(ε)ε−2V(⋅−y/ε) λ(ε) being a real-valued, C∞ [0,1] function with λ(0)=1 and V∈L ∞ is such that supp V is contained in the Wigner–Seitz cell. By using the direct integral decomposition, we reduce the problem to the convergence of the reduced Hamiltonian Hε (θ)=−Δθ +λ(ε)ε−2V(⋅/ε). In order to find the limit when ε→0+ of [Hε(θ)−E]−1, we also study the properties of its integral kernel.