The limiting absorption principle and spectral theory for steady-state wave propagation in inhomogeneous anisotropic media

Abstract

Many wave propagation phenomena of classical physics are governed by systems of the Schrödinger form-iD _t u+Λu=f(x,t) where

$$\Lambda = - iE(x)^{ - 1} \sum\limits_{j = 1}^n {(A_j D_j )} $$

(1)

, (1) E(x) and the A _j are Hermitian matrices, E(x) is positive definite and the A_j are constants. If f(x, t)=e ^−iλt f(x) then a corresponding steady-state solution has the form u(x, t)=e^{−i λ t}ν(x) where ν(x) satisfies (Λ-λ) ν=f(x), xεR ⁿ. (2) This equation does not have a unique solution for λεR ¹−{0} and it is necessary to add a radiation condition for ¦ x ¦ → ∞ which ensures that ν(X) behaves like an outgoing wave.

The limiting absorption principle provides one way to construct the correct outgoing solution of (2). It is based on the fact that Λ defines a self-adjoint operator on the Hilbert space ℋ defined by the energy inner product

$$(u,v) = \int\limits_{R^n } {u^* } E{\text{ }}v{\text{ }}d{\text{ }}x$$

(2)

. It follows that if ζ=λ+iσ and σ≠0 then (Λ-ζ) ν=f has a unique solution

$$v(,\zeta ) = R_\zeta (\Lambda )f \in $$

(3)

ℋ where

$$R_\zeta (\Lambda ) = (\Lambda - \zeta )^{ - 1} $$

(4)

is the resolvent for Λ on ℋ. The limiting absorption principle states that

$$v(,\lambda ) = \mathop {\lim }\limits_{\sigma \to 0} v(,\lambda + i\sigma )$$

(5)

(3) exists, locally on R ⁿ, and defines the outgoing solution of (2).

This paper presents a proof of the limiting absorption principle, under suitable hypotheses on E(x) and the A _j . The proof is based on a uniqueness theorem for the steady-state problem and a coerciveness theorem for nonelliptic operators Λ of the form (1) which were recently proved by the authors.

The coerciveness theorem and limiting absorption principle also provide information about the spectrum of Λ. It is proved in this paper that the point spectrum of Λ is discrete (that is, there are finitely many eigenvalues in any interval) and that the continuous spectrum of Λ is absolutely continuous.