From Classical monodromy to Spectral monodromy

Thời gian: 10:00 đến 11:30 ngày 16/11/2016, 10:00 đến 11:30 ngày 23/11/2016, 10:00 đến 11:30 ngày 30/11/2016,

Địa điểm:

Tóm tắt:

In the classical theory, classical monodromy is defined for integrable Hamiltonian systems on symplectic manifolds as a topological invariant that obstructs the existence of global action-angle coordinates on the phase space.
Dually, quantum monodromy is completely defined in the joint spectrum of commuting selfadjoint operators, in the sense of the semiclassical limit.

Whether a monodromy can be defined for only one semiclassical operator? That is how to detect the modification of action-angle variables from only one  spectrum? Spectral monodromy, defined directly from the spectrum of a single non-selfadjoint pseudodifferential operator (with two degrees of freedom) allows to response this question. Moreover this monodromy allows to recover the monodromy of the underlying classical system.

In this talk, we will present:
Part 1:
- the classical monodromy, the quantum monodromy;
- the monodromy of normal operators.

Part 2:
the spectral monodromy of small non-selfadjoint perturbations of a selfadjoint quantum Hamiltonian, assuming that the principal symbol of the selfadjoint unperturbed part is in two cases:
- a completely integrable system;
- a nearly integrable system, with a globally non-degenerate condition.