Microlocal properties of scattering matrices

Abstract

We consider scattering theory for a pair of operators H 0 and H = H 0 + V on L 2 (M, m), where M is a Riemannian manifold, H 0 is a multiplication operator on M and V is a pseudodifferential operator of order −µ, µ > 1. We show that a time-dependent scattering theory can be constructed, and the scattering matrix is a pseudodifferential operator on each energy surface. Moreover, the principal symbol of the scattering matrix is given by a Born approximation type function. The main motivation of the study comes from applications to discrete Schrödigner operators, but it also applies to various differential operators with constant coefficients and short-range perturbations on Euclidean spaces.Let I be a compact interval and we assumeand p −1 0 (I) is compact. We now consider the scattering theory for the pair (H, H 0 ) on the energy interval I, i.e., we study the absolutely continuous spectrum of H on I. We denote the spectral projection of an operator A on J ⊂ R by E J (A). Then the wave operatorsexist and they are complete: Ran W I ± = E I (H)H ac (H). Moreover, the point spectrum σ(H) ∩ I is finite including the multiplicities (see Section 2).We write the energy surface of H 0 with an energy λ ∈ I byΣ λ is a regular submanifold in M , and we let m λ be the smooth density on Σ λ characterized as follows: m λ = i * m λ , wherem λ ∈ d−1 (M ) such thatm λ ∧ dp 0 = m, and i : Σ λ ֒→ M is the embedding. (Note m λ is uniquely determined whereasm λ is not.) The scattering operator is defined by S I = (W I + ) * W I − , H → H, and it commutes with H 0 . Hence S I is decomposed to a family of operators {S(λ)} λ∈I , where S(λ) is a unitary operator on L 2 (Σ λ , m λ ) for a.e. λ ∈ I. S(λ) is called the scattering matrix (see Section 5 for the detail). Our main result is the following: Theorem 1.

Microlocal resolvent estimates, revisited

“…We suppose p −1 0 (I) = ξ ∈ M p 0 (ξ) ∈ I is compact, and v(ξ) = 0 for ξ ∈ p −1 0 (I), i.e., I does not contain critical values of p 0 . Under this assumption, it is easy to see that the following claims using the standard Mourre theory (see, e.g., [11], [1], [15] Section 2): σ p (H) ∩ I is discrete, each eigenvalues are finite dimensional, and for λ ∈ I \ σ p (H), s > 1/2, the limits…”

“…Then we can apply Theorem 2.1 to H = H 0 + V . We refer Nakamura [15] Section 7 for the detail of the construction.…”

“…Remark 1.2. A mocrolocal resolvent estimate of this form was proved in [15] for the short range case (in more general setting as in Section 2), and applied to the analysis of scattering matrices. The proof relies on a construction of parametrices.…”

Long-Range Scattering for Discrete Schrödinger Operators

“…Under Assumptions 1.1 and 1.3, the singular continuous spectrum of H is empty (see, e.g., [12]). In the following, we write V for Ṽ without confusion.…”

“…Discrete Schrödinger operator describes the state of electrons in solid matters with graph structure. Spectral properties of discrete Schrödinger operators have been studied in [2], [4], [7], [11], [12], [14].…”