Tunnel effect and symmetries for non-selfadjoint operators

Hitrik, Michael

Journées équations aux dérivées partielles (2013), article no. 5, 12 p.

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 $\mathrm{𝒫𝒯}$-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, but when the temperatures of the baths are different, we show that the supersymmetric approach may break down. We also discuss $\mathrm{𝒫𝒯}$–symmetric quadratic differential operators with real spectrum and characterize those that are similar to selfadjoint operators. This talk is based on joint works with Emanuela Caliceti, Sandro Graffi, Frédéric Hérau, and Johannes Sjöstrand.

Classification : 35P15, 35P20, 47A10, 81Q20, 81Q60, 82C31
Mots clés : Non-selfadjoint, supersymmetry, Kramers-Fokker-Planck, tunneling, exponentially small eigenvalue, chain of oscillators, semiclassical limit, $\mathcal{PT}$–symmetry, quadratic operator, Hamilton map

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