Spectral and scattering properties at thresholds for the Laplacian in a half-space with a periodic boundary condition

Abstract

For the scattering system given by the Laplacian in a half-space with a periodic boundary condition, we derive resolvent expansions at embedded thresholds, we prove the continuity of the scattering matrix, and we establish new formulas for the wave operators.

Discrete Laplacian in a half-space with a periodic surface potential I: Resolvent expansions, scattering matrix, and wave operators

“…Our proof is based on two lemmas. The first lemma complements the continuity properties established in Section 4, and it is similar to Lemma 5.3 of [16] in the continuous setting: Lemma 5.5. For any θ ∈ [0, 2π] and j, j ∈ {1, .…”

“…. , N}, while in [16] the scattering channels also open at specific energies but do no close before reaching infinity.…”

“…We are now ready to derive resolvent expansions for the fibered Hamiltonians H θ using the inversion formulas and iterative scheme developed and used in [15,16]. For that purpose, we set…”

“…The iterative procedure stops here, for the same reason as the one presented in the proof of [16,Prop. 3.3] once we observe that…”

“…We establish in this section continuity properties of the channel scattering matrices for the pair (H θ , H θ 0 ) for each θ ∈ [0, 2π]. Our approach is similar to that of [16,Sec. 4], with one major difference: Here, the scattering channels open at energies λ θ j − 2 and close at energies λ θ j + 2 for j ∈ {1, .…”