Rational invariant tori and band edge spectra for non-selfadjoint operators

Abstract

We study semiclassical asymptotics for spectra of non-selfadjoint perturbations of selfadjoint analytic h-pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part is completely integrable. Complete asymptotic expansions are established for all individual eigenvalues in suitable regions of the complex spectral plane, near the edges of the spectral band, coming from rational flow-invariant Lagrangian tori.

Subelliptic resolvent estimates for non-self-adjoint semiclassical Schrödinger operators

“…Theirs uses the FBI transform in a compact region of phase space and and Weyl-Hörmander calculus with a suitable metric near infinity, while ours works globally using the Wick quantization and some standard Weyl calculus. Hitrik and Sjöstrand attained a similar estimate for certain one-dimensional non-self-adjoint Schrödinger operators [7], with ellipticity assumptions on the potential. Also, Dencker, Sjöstrand, and Zworski showed that for non-self-adjoint semiclassical operators, under suitable assumptions including ellipticity at infinity, the resolvent can be similarly estimated in a small region near a boundary point of the range of the symbol, away from critical values of the symbol [3].…”

“…These settings can range physical problems to purely mathematical ones. Such examples include the study of the Ginzburg-Landau equation in superconductivity [1], [4], the Orr-Somerfeld operator in fluid dynamics [11], [12], the theory of scattering resonances [14], or non-self-adjoint perturbations of self-adjoint operators [7]. In the self-adjoint case, the spectral theorem provides a powerful tool to control the resolvent of Schrödinger operators.…”

Asymptotic Lattices, Good Labellings, and the Rotation Number for Quantum Integrable Systems

“…On the other hand, if we consider the good labelling k , the translation invariance property (24) gives similarly j) .…”

“…The following lemma shows that this process is uniformly well-defined if these points lie in ball of size O( ) and 0 is small enough. Then, Step (iii) amounts to picking up the natural parallel transport defined in (24):…”

“…For quasi-periodic dynamics, this idea was exploited to recover the Birkhoff normal form from the complex spectrum in [21]. A strong motivation for our work is the recent paper [24], which naturally leads to this intriguing question: can you detect the rationality of the rotation number from the spectrum?…”