Semiclassical Inverse Spectral Theory for Singularities of Focus–Focus Type

Semiclassical Inverse Spectral Theory for Singularities of Focus-focus Type

We prove, assuming that the Bohr-Sommerfeld rules hold, that the joint spectrum near a focus-focus critical value of a quantum integrable system determines the classical Lagrangian foliation around the full focus-focus leaf. The result applies, for instance, to ~-pseudodifferential operators, and to Berezin-Toeplitz operators on prequantizable compact symplectic manifolds.

Inverse Spectral Theory for One

Symplectic inverse spectral theory for pseudodifferential operators

We prove, under some generic assumptions, that the semiclassical spectrum modulo O(~) of a one dimensional pseudodifferential operator completely determines the symplectic geometry of the underlying classical system. In particular, the spectrum determines the hamiltonian dynamics of the principal symbol.

Semiclassical singularities from bifurcating orbits.

We investigated numerically, for a generic quantum system (a kicked top), how the singular behavior of classical systems at bifurcations is reflected by their quantum counterpart. Good agreement is found with semiclassical predictions.

Resummation and the semiclassical theory of spectral statistics

We address the question as to why, in the semiclassical limit, classically chaotic systems generically exhibit universal quantum spectral statistics coincident with those of random-matrix theory. To do so, we use a semiclassical resummation formalism that explicitly preserves the unitarity of the quantum time evolution by incorporating duality relations between short and long classical orbits. ...