78 research outputs found
Rational invariant tori, phase space tunneling, and spectra for non-selfadjoint operators in dimension 2
We study spectral asymptotics and resolvent bounds for non-selfadjoint
perturbations of selfadjoint -pseudodifferential operators in dimension 2,
assuming that the classical flow of the unperturbed part is completely
integrable. Spectral contributions coming from rational invariant Lagrangian
tori are analyzed. Estimating the tunnel effect between strongly irrational
(Diophantine) and rational tori, we obtain an accurate description of the
spectrum in a suitable window in the complex spectral plane, provided that the
strength of the non-selfadjoint perturbation (or sometimes )
is not too large.Comment: 66 page
Tunnel effect for Kramers-Fokker-Planck type operators: return to equilibrium and applications
In the first part of this work, we consider second order supersymmetric
differential operators in the semiclassical limit, including the
Kramers-Fokker-Planck operator, such that the exponent of the associated
Maxwellian is a Morse function with two local minima and one saddle
point. Under suitable additional assumptions of dynamical nature, we establish
the long time convergence to the equilibrium for the associated heat semigroup,
with the rate given by the first non-vanishing, exponentially small,
eigenvalue. In the second part of the paper, we consider the case when the
function has precisely one local minimum and one saddle point. We also
discuss further examples of supersymmetric operators, including the Witten
Laplacian and the infinitesimal generator for the time evolution of a chain of
classical anharmonic oscillators
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