Resolvent estimates and local energy decay for hyperbolic equations

Abstract

We examine the cut-off resolventχ, where Δ D is the Laplacian with Dirichlet boundary condition and χ ∈ C ∞ 0 (R n ) equal to 1 in a neighborhood of the obstacle K. We show that if R χ (λ ) has no poles forThis estimate implies a local energy decay. We study the spectrum of the Lax-Phillips semigroup Z(t) for trapping obstacles having at least one trapped ray.

Wave Decay on Convex Co-Compact Hyperbolic Manifolds

“…Using Proposition 2.1, this implies that the resolvent R(λ) is holomorphic in a similar set (possibly by taking ǫ > 0 smaller). Then an easy consequence of the maximum principle as in [38,2] together with a rough exponential bound for the resolvent allows to get a polynomial bound for ||χ 1 R(λ)χ 2 || on the {Re(λ) = δ; λ = δ}. Corollary 2.6.…”

Estimates of holomorphic functions in zero-free domains

“…In [1] this conjecture has been proved for n = 3 using a reduction to a semiclassical Schrödinger operator and a suitable estimate for the resolvent of a complex scaling operator. For dimensions n > 3 the result in [1] seems to be not optimal since we may deduce only a bound…”

“…. , N and we search f1 (z) in the form f 1 (z) = N k=1 c k (z)e k .For the functions c k (z) we get a linear systemc k (z) + N j=1 c j (z)(B(z)e j , e k ) = (h(z), e k ) = h k (z), k = 1, . .…”

Propagation through trapped sets and semiclassical resolvent estimates

“…See [12,Theorem 6.1] for (1.2) in the asymptotically hyperbolic case, and see [11] and [30,Corollary 1] for the asymptotically conic case. Bony and Petkov [2] prove (1.2) for a general "black box" perturbation of the Laplacian in R n assuming only that there is a resonance-free strip, and it is likely that this condition suffices for asymptotically conic or hyperbolic manifolds as well.…”