Spectral projection, residue of the scattering amplitude and Schrödinger group expansion for barrier-top resonances

Quasi-Normal Modes and Exponential Energy Decay for the Kerr-de Sitter Black Hole

“…The third case in Proposition 7.3 is where trapping occurs, and we analyse it as in [38]: (See also [10] for a different method of solving the same problem.) Proposition 7.5.…”

Resonances for homoclinic trapped sets

“…This justifies that neighborhoods of these points (or of Γ(h) for technical reasons) are removed from the set (4.12). Note also that the polynomial resolvent estimate in the complex plane has already been obtained by Michel and the first author [11,Theorem 6.4] and in a previous paper [10,Theorem 3.1]. Of course, the discussions of Section 4.2 are pointless here since P has no accumulation curve.…”

“…This explains that the first accumulation curves have a horizontal asymptote as Re σ → +∞. That the two accumulation curves coincide could be explained by the non-vanishing of the symbol of the resonant states on the curves γ • (see (7.11)) which is typically a property that is true only for the first line of resonances generated by hyperbolic trappings (see Theorem 4.1 iv) of [10] for barrier-top resonances).…”

“…This result has also been proved by Briet, Combes and Duclos [13,14] (see [79] in dimension 1). The interested reader may find in [10] a precise description of the spectral projection associated to these resonances.…”

Two Minicourses on Analytic Microlocal Analysis

“…In the theory of scattering poles (resonances) and other branches of spectral theory for non-self-adjoint (pseudo-)differential operators, many works rely on phase space deformations which are now global. Since this activity started later we simply refer to some of the works which also include some of those devoted to other global questions: [37]- [66].…”