Diophantine tori and non-selfadjoint inverse spectral problems

Abstract

Abstract. We study a semiclassical inverse spectral problem based on a spectral asymptotics result of [13], which applies to small non-selfadjoint perturbations of selfadjoint h-pseudodifferential operators in dimension 2. The eigenvalues in a suitable complex window have an expansion in terms of a quantum Birkhoff normal form (QBNF) for the operator near several Lagrangian tori that are invariant under the classical dynamics and satisfy a Diophantine condition. In this work, we prove that the normal form near a single Diophantine torus is uniquely determined by the associated eigenvalues. We also discuss the normalization procedure and symmetries of the QBNF near a Diophantine torus.

Inverse spectral theory for semiclassical Jaynes–Cummings systems

“…The problem treated in this paper belongs to a class of semiclassical inverse spectral questions which has attracted much attention in recent years, e.g. [21,24,25,15,29,34,39], which goes back to pioneer works of Bérard [1], Brüning-Heintze [3], Colin de Verdière [12,13], Duistermaat-Guillemin [19], and Guillemin-Sternberg [22], in the 1970s/1980s, and are closely related to inverse problems that are not directly semiclassical but do use similar microlocal techniques for some integrable systems, as in [40] (see also [41] and references therein).…”

Symplectic invariants of semitoric systems and the inverse problem for quantum systems

“…The inverse spectral problem for quantum integrable systems. The inverse problem we are going to discuss next belongs to a class of semiclassical inverse spectral questions which has been the focus of intense attention in recent years [26,51,52,66,84,89,108]. The problem goes back to pioneer works of Bérard [10], Brüning-Heintze [15], Colin de Verdière [24,25], Duistermaat-Guillemin [35], and Guillemin-Sternberg [48], in the 1970s/1980s, and is closely related to inverse problems that are not directly semiclassical but use similar microlocal methods [115,116].…”

Quantum footprints of Liouville integrable systems

“…, P n , when the phase space M is 2n-dimensional. In fact, even for operators that are not quantum integrable but still have a completely integrable classical limit, quite precise results can be obtained, for both direct and inverse problems; see [16,15], and references therein.…”