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The radiation condition at infinity for the high-frequency Helmholtz equation with source term: a wave packet approach

François Castella (2004)

Journées Équations aux dérivées partielles

We consider the high-frequency Helmholtz equation with a given source term, and a small absorption parameter α > 0 . The high-frequency (or: semi-classical) parameter is ε > 0 . We let ε and α go to zero simultaneously. We assume that the zero energy is non-trapping for the underlying classical flow. We also assume that the classical trajectories starting from the origin satisfy a transversality condition, a generic assumption.Under these assumptions, we prove that the solution u ε radiates in the outgoing...

Tunnel effect and symmetries for non-selfadjoint operators

Michael Hitrik (2013)

Journées Équations aux dérivées partielles

We study low lying eigenvalues for non-selfadjoint semiclassical differential operators, where symmetries play an important role. In the case of the Kramers-Fokker-Planck operator, we show how the presence of certain supersymmetric and 𝒫𝒯 -symmetric structures leads to precise results concerning the reality and the size of the exponentially small eigenvalues in the semiclassical (here the low temperature) limit. This analysis also applies sometimes to chains of oscillators coupled to two heat baths,...

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