We study the spectrum of the semiclassical Witten Laplacian associated to a smooth function on . We assume that is a confining Morse–Bott function. Under this assumption we show that admits exponentially small eigenvalues separated from the rest of the spectrum. Moreover, we establish Eyring–Kramers formula for these eigenvalues. Our approach is based on microlocal constructions of quasimodes near the critical submanifolds.
Nous étudions le spectre du Laplacien de Witten semi-classique associé à une fonction lisse sur . Nous supposons que est une fonction de Morse–Bott confinante. Sous cette hypothèse, nous montrons que admet des valeurs propres exponentiellement petites séparées du reste du spectre. De plus, nous établissons une formule d’Eyring–Kramers pour ces valeurs propres. Notre approche est basée sur des constructions microlocales de quasimodes au voisinage des sous-variétés critiques.
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Keywords: Semiclassical analysis, Spectral asymptotic, Witten Laplacian, Overdamped Langevin dynamics.
Mot clés : Analyse semiclassique, asymptotique spectrale, laplacien de Witten, dynamique de Langevin suramortie.
Assal, Marouane 1; Bony, Jean-François 2; Michel, Laurent 2
@unpublished{AIF_0__0_0_A91_0, author = {Assal, Marouane and Bony, Jean-Fran\c{c}ois and Michel, Laurent}, title = {Metastable diffusions with degenerate drifts}, journal = {Annales de l'Institut Fourier}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, year = {2024}, doi = {10.5802/aif.3636}, language = {en}, note = {Online first}, }
Assal, Marouane; Bony, Jean-François; Michel, Laurent. Metastable diffusions with degenerate drifts. Annales de l'Institut Fourier, Online first, 33 p.
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