Rational invariant tori, phase space tunneling, and spectra for non-selfadjoint operators in dimension 2

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

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.

Diophantine tori and Weyl laws for non-selfadjoint operators in dimension two

We study the distribution of eigenvalues for non-selfadjoint perturbations of selfadjoint semiclassical analytic pseudodifferential operators in dimension two, assuming that the classical flow of the unperturbed part is completely integrable. An asymptotic formula of Weyl type for the number of eigenvalues in a spectral band, bounded from above and from below by levels corresponding to Diophantine invariant Lagrangian tori, is established. The Weyl law is given in terms of the long time averages of the leading non-selfadjoint perturbation along the classical flow of the unperturbed part.

Diophantine tori and spectral asymptotics for non-selfadjoint operators

We study spectral asymptotics for small non-selfadjoint perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part possesses several invariant Lagrangian tori enjoying a Diophantine property. We get complete asymptotic expansions for all eigenvalues in certain rectangles in the complex plane in two different cases: in the first case, we assume that the strength $ε$ of the perturbation is ${\cal O}(h^δ)$ for some $δ>0$ and is bounded from below by a fixed positive power of $h$. In the second case, $ε$ is assumed to be s…

无序隧道势的扭曲双层石墨烯中无小魔角的缺失

Absence of small magic angles for disordered tunneling potentials in twisted bilayer graphene

本研究通过引入小随机扰动，利用Sjöstrand方法证明在足够小的扭曲角下，Bistritzer-MacDonald哈密顿量以压倒性概率不呈现平坦带，从而不存在所谓的魔角，并证明了手性极限下特征值的概率Weyl定律。

Construction of quasimodes for non-selfadjoint operators via propagation of Hagedorn wave-packets

We construct quasimodes for some non-selfadjoint semiclassical operators at the boundary of the pseudo-spectrum using propagation of Hagedorn wave-packets. Assuming that the imaginary part of the principal symbol of the operator is non-negative and vanishes on certain points of the phase-space satisfying a subelliptic finite-type condition, we construct quasimodes that concentrate on these non-damped points. More generally, we apply this technique to construct quasimodes for non-selfadjoint semiclassical perturbations of the harmonic oscillator that concentrate on non-damped periodic orbits or…

Diophantine tori and nonselfadjoint inverse spectral problems

We study a semiclassical inverse spectral problem based on a spectral asymptotics result of arXiv:math/0502032, 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 for the operator near several Lagrangian tori which 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 a…