On the location of resonances for Schrödinger operators in the semiclassical limit I. Resonances free domains

Distribution of resonances for spherical black holes

“…The purpose of this note is to apply the methods of geometric scattering theory developed by Briet-Combes-Duclos [6], Gérard-Sjöstrand [14], Mazzeo-Melrose [22] and the second author [30] in the simplest model of a Black Hole: the De SitterSchwarzschild metric. We show that the resonances (or the quasi normal modes, in the terminology of Chandrasekhar [8]) are globally defined in C and that in a strip below the real axis and for large angular momenta, l, they are well approximated by the "pseudo-poles"…”

“…The general phenomenon of resonances associated to simple hyperbolic orbits is now well known and was studied in [5,19,13,6,14,15]. Ikawa [19] and Gérard [13] showed that in scattering by two strictly convex bodies the unique closed trajectory (which is hyperbolic: it is given by the ray refelected at the points of closest distance between the two bodies) generates a lattice of resonances.…”

“…It is organized as follows: In Sect.2 we show how to apply the results of [22] in the De Sitter setting and its perturbations (only the exact Laplace-Beltrami operator was explicitly discussed in that paper) and in Sect.3 we review the geometry of trajectories of the classical flow of P and in particular the structure of the trapped set. Finally in Sect.4 we will separate variables and apply the results of [6,14,27,30] in the resulting one dimensional problem.…”

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

“…In any dimension, another approach is perhaps also possible. One may first try to calculate the resonant state f with various methods (using, for example, the works of Briet, Combes and Duclos [5], Sjöstrand [37] or Hassell, Melrose and Vasy [19]). It then remains to calculate the constant c. This question is equivalent to the calculation of the scalar product (f, f ) = f 2 .…”

“…In particular, we show quantitatively to what extent the presence of these resonances drives the behavior of the scattering amplitude and of the Schrödinger group. The resonances generated by the maximum point, supposed to be nondegenerate, of the potential (usually called barrier-top resonances) have been studied by Briet, Combes and Duclos [5,6] and Sjöstrand [37]. These authors have given a precise description of the set Res(P ) = {z α ≈ E 0 − ih n j=1 α j + 1 2 λ j , α ∈ N n } of resonances in any disc of size h centered at the maximum value E 0 of the potential.…”