Quantum-mechanical scattering in exterior domains with impenetrable periodic boundaries and short-range potentials

Abstract

We study the scattering of a nonrelativistic particle in an exterior domain (=open connected subset) Ω⊂Rν(ν⩾2) containing a half-space and contained in another half-space, and having an impenetrable periodic boundary ∂Ω. “Impenetrable” means that (generalized) homogeneous Dirichlet conditions are imposed on ∂Ω. We prove the existence and completeness of the wave operators W±=limt→±∞ exp(itH1)P exp(−itH0) corresponding to the scattering of a nonrelativistic particle in Ω by the combined effect of the boundary and a short-range potential present in Ω. Here H0=−Δ is the negative distributional Laplacian in the Hilbert space H0=L2(Rν), H1=−ΔD(Ω)+V, ΔD(Ω) being the Dirichlet Laplacian in the Hilbert space H=L2(Ω), V an operator of multiplication in ℋ by a bounded measurable function V(x) on Ω having the periodicity of the boundary, and P:H0→H an identification operator. The operators W± model the quantum-mechanical scattering of low-energy atoms by crystal surfaces, with V modeling the interaction between the incident particles and the surface atoms. This interaction is idealized by assuming that V(x) depends solely on xν when xν>a, a being a sufficiently large positive constant, and xν the component of x∈Rν directed perpendicularly to the surfaces of the above two half-spaces. Under this and other hypotheses on Ω and V stated precisely in the paper, we prove that W± exist as partially isometric operators whose initial sets have a transparent physical meaning. Moreover, we prove the following: (a) Ran W±=Hscatt; and (b) W± are asymptotically complete, in the sense that H=Hscatt⊕Hsurf. Here Hscatt and Hsurf are suitably defined subspaces of scattering and surface states of ℋ. These results are proved by using direct-integral techniques, asymptotic methods from the theory of ODEs, and methods analogous to those of Lyford. The present paper generalizes an earlier one by the author for the case V=0.