Tunnel effect and symmetries for non-selfadjoint operators

Michael Hitrik[1]

  • [1] Department of Mathematiscs University of California, Los Angeles Los Angeles, CA 90095–1555, United States

Journées Équations aux dérivées partielles (2013)

  • page 1-12
  • ISSN: 0752-0360

Abstract

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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, but when the temperatures of the baths are different, we show that the supersymmetric approach may break down. We also discuss 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.

How to cite

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Hitrik, Michael. "Tunnel effect and symmetries for non-selfadjoint operators." Journées Équations aux dérivées partielles (2013): 1-12. <http://eudml.org/doc/275439>.

@article{Hitrik2013,
abstract = {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 $\mathcal\{PT\}$-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 $\mathcal\{PT\}$–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.},
affiliation = {Department of Mathematiscs University of California, Los Angeles Los Angeles, CA 90095–1555, United States},
author = {Hitrik, Michael},
journal = {Journées Équations aux dérivées partielles},
keywords = {Non-selfadjoint; supersymmetry; Kramers-Fokker-Planck; tunneling; exponentially small eigenvalue; chain of oscillators; semiclassical limit; $\mathcal\{PT\}$–symmetry; quadratic operator; Hamilton map},
language = {eng},
pages = {1-12},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Tunnel effect and symmetries for non-selfadjoint operators},
url = {http://eudml.org/doc/275439},
year = {2013},
}

TY - JOUR
AU - Hitrik, Michael
TI - Tunnel effect and symmetries for non-selfadjoint operators
JO - Journées Équations aux dérivées partielles
PY - 2013
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 12
AB - 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 $\mathcal{PT}$-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 $\mathcal{PT}$–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.
LA - eng
KW - Non-selfadjoint; supersymmetry; Kramers-Fokker-Planck; tunneling; exponentially small eigenvalue; chain of oscillators; semiclassical limit; $\mathcal{PT}$–symmetry; quadratic operator; Hamilton map
UR - http://eudml.org/doc/275439
ER -

References

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