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

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.

非自伴算子的丢番图环面与谱渐近性

Diophantine tori and spectral asymptotics for non-selfadjoint operators

我们研究了2维自伴$h$ -伪微分算子的非自伴小扰动的谱渐近性，假设未扰动部分的经典流具有几个不变的拉格朗日环面，并具有丢番图性质.在两种不同的情况下，我们得到了复平面中某些矩形中的所有特征值的完全渐近展开式：在第一种情况下，我们假设扰动的强度$ε$对于某些$δ>0$是${\cal O}(h^δ)$，并且从下面有界于$h$的固定正幂。在第二种情况下，$ε$被假定为足够小，但独立的$h$，我们描述的特征值完全在一个固定的$h$独立域的复谱平面。

二维非自伴算子的有理不变式环面、相空间隧穿和谱

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

在经典流完全可积的条件下，研究了二维自伴$h$ -伪微分算子非自伴扰动的谱渐近性和预解式界.分析了来自有理不变式拉格朗日环面的谱贡献。通过估计强无理（丢番图）和有理环面之间的隧道效应，在复谱平面上的一个合适的窗口中，只要非自伴扰动$\gg h$（有时也称$\gg h^2$）的强度不太大，就可以得到谱的精确描述.

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…

Skew-selfadjoint Dirac systems with rational rectangular Weyl functions: explicit solutions of direct and inverse problems and integrable wave equations

In this paper we study direct and inverse problems for discrete and continuous time skew-selfadjoint Dirac systems with rectangular (possibly non-square) pseudo-exponential potentials. For such a system the Weyl function is a strictly proper rational rectangular matrix function and any strictly proper rational matrix function appears in this way. In fact, extending earlier results, given a strictly proper rational matrix function we present an explicit procedure to recover the corresponding potential using techniques from mathematical system and control theory. We also introduce and study a no…

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…