科研之友
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摘要
Let H(0) be a self-adjoint operator in some Hilbert space H, and let lambda(0) be a (possibly degenerate) eigenvalue of H(0) embedded in its essential spectrum sigma(ess)(H(0)) with corresponding eigenprojection Pi(0). For small |kappa|, let H(kappa) be a family of perturbed Hamiltonians, which is analytic in a generalized Balslev-Combes sense. Following Hunziker's approach in (Commun Math Phys 132: 177-188, 1990), we discuss the corrections to exponential decay in
P(0)e (-itH(kappa)) g(II(kappa)) Pi(0) - D(kappa)e (-ith(kappa)) D(kappa) + R(kappa, t),
where D(kappa) = Pi(0) + O(kappa(2)) (kappa -> 0) and h(kappa) is some family of in general non self-adjoint bounded operators with Ranh(kappa) = Ran Pi(0), leaving Ran Pi(0) invariant, and 0 <= g <= 1 is a cut-off function with g(lambda(0)) = 1 and sufficiently small support. Our main result is a sharp estimate of the remainder R(kappa, t) in terms of the Gevrey index a > 1, b > 0 of g is an element of Gamma(a, b) (R):
parallel to R(kappa, t)parallel to <= = O(kappa(2)) e(-Ct1/a) (t >= 0, C < ab(-1/a) ,kappa -> 0).
- 出版日期2010-8
