Resolvent Estimates and Scattering States for Dissipative Systems

result is Theorem 1. Let A0 and B be self-adjoint operators in H. Assume that B is non-negative, D(A0) ⊂ D(B) and (A1) ∼ (A3). Then (1) A has no real eigenvalues. (2) The wave operator W = s− lim t→∞U0(−t)V (t) exists. Moreover W is not zero as an operator in H. We shall use (1.5) to get (1.6) or (1.7) (see Section 3). Theorem 1 (2) implies that there exist scattering states of dV (t)g/dt = −iAV (t)g, g ∈ D(A) as follows: Corollary 2. Let A0 and B be self-adjoint operators in H. Assume that B is non-negative, D(A0) ⊂ D(B) and (A1) ∼ (A3). Then there exist non-trivial initial data f ∈ H and f+ ∈ H such that for any k = 0, 1, 2, · · · , and ζ0 ∈ C satisfying ζ0 > 0 lim t→∞ ‖V (t)(A− ζ0) −kf − U0(t)(A0 − ζ0)−kf+‖H = 0. The proof of Theorem 1 and Corollary 2 will be given in Section 2. Abstractly, Mochizuki’s result [17] can be explained as follows. Let A0 and B be self-adjoint operators in H. If we suppose that B is non-negative, bounded and there exist positive constants C and η such that sup 0<|Imζ|<η ‖ √ B(A0 − ζ)−1 √ B‖ C (1.8) instead of (A1), (A2) and (A3), we can prove the same conclusion as in Theorem 1 (see Lemma 2.1 of Section 2). Dissipative Scattering 195