Asymptotics of the number of Rayleigh resonances

From Semiclassical Strichartz Estimates to Uniform $L^p$ Resolvent Estimates on Compact Manifolds

“…Let ϕ, ϕ, ψ ∈ C ∞ 0 (R n ) be such that ϕ = 1 near supp (ϕ), and ψ = 1 near supp ( ϕ). Let g be a C ∞ Riemannian metric on R n such that ∂ α x g ∈ L ∞ for all α, and let Furthermore, 24) see (3.13). Let B ∈ Op w h S −N (R 2n ) for all N , and let χ = χ(x, ξ; h) ∈ S −N (R 2n ) for all N be such that…”

“…Spectral region III. Let τ ∈ C be such that |Re τ | ≤ 1 2 |Im τ | and |Im τ | sufficiently large, depending on the damping coefficient a. Multiplying (5.1) by u, integrating by parts and taking the real part, we get 24) where dV g is the Riemannian volume element of M . Using that |Re τ | ≤ 1 2 |Im τ |, we obtain that…”

Rayleigh-type surface quasimodes in general linear elasticity

“…Rayleigh waves have been studied in several papers with the emphasis of getting information about resonances in scattering theory [Stefanov and Vodev 1994;1995;1996;Sjöstrand and Vodev 1997;Vodev 1997;Stefanov 2000], and, for anisotropic media, [Kawashita and Nakamura 2000]. Stefanov [2000] uses Rayleigh quasimodes to derive lower bounds on the number of resonances.…”

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“…Rayleigh waves have been studied in several papers with the emphasis of getting information about resonances in scattering theory [Stefanov and Vodev 1994;1996;Sjöstrand and Vodev 1997;Vodev 1997;Stefanov 2000], and, for anisotropic media, [Kawashita and Nakamura 2000]. Stefanov [2000] uses Rayleigh quasimodes to derive lower bounds on the number of resonances.…”