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Article Dans Une Revue Journal of Mathematical Physics Année : 2014

Spectral monodromy of non selfadjoint operators

Résumé

We propose to build a combinatorial invariant, called the spectral monodromy, from the spectrum of a single non-selfadjoint h-pseudodifferential operator with two degrees of freedom in the semi-classical limit. Our inspiration comes from the quantum monodromy defined for the joint spectrum of an integrable system of n commuting selfadjoint h-pseudodifferential operators, given by S. Vu Ngoc. The first simple case that we treat in this work is a normal operator. In this case, the discrete spectrum can be identified with the joint spectrum of an integrable quantum system. The second more complex case we propose is a small perturbation of a selfadjoint operator with a classical integrability property. We show that the discrete spectrum (in a small band around the real axis) also has a combinatorial monodromy. The difficulty here is that we do not know the description of the spectrum everywhere, but only in a Cantor type set. In addition, we also show that the monodromy can be identified with the classical monodromy defined by J. Duistermaat.
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Dates et versions

hal-00766411 , version 1 (18-12-2012)
hal-00766411 , version 2 (06-03-2013)

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Quang Sang Phan. Spectral monodromy of non selfadjoint operators. Journal of Mathematical Physics, 2014, 55, ⟨10.1063/1.4855475⟩. ⟨hal-00766411v2⟩
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