Singular Bohr-Sommerfeld Rules for 2D Integrable Systems

Singular Bohr-Sommerfeld conditions for 1D Toeplitz operators: hyperbolic case

In this article, we state the Bohr-Sommerfeld conditions around a singular value of hyperbolic type of the principal symbol of a self-adjoint semiclassical Toeplitz operator on a compact connected Kähler surface. These conditions allow the description of the spectrum of the operator in a fixed size neighbourhood of the singularity. We provide numerical computations for three examples, each associated to a different topology.

Bohr-Sommerfeld quantization conditions for non-selfadjoint perturbations of selfadjoint operators in dimension one

In this paper, we give a description of the spectrum of a class of non-selfadjoint perturbations of selfadjoint operators in dimension one and we show that it is given by Bohr-Sommerfeld quantization conditions. To achieve this, we make use of previous work by Michael Hitrik, Anders Melin and Johannes Sj{ö}strand. We also give an application of our result in the case of PT-symmetric pseudo-differential operators.

玻尔-索默菲尔德规则对于系统

Bohr--Sommerfeld rules for systems

本文提出了一种完整的自洽形式，用于描述实线上由相空间简单闭合曲线产生的半经典自共轭2×2系统中的玻尔-索默菲尔德量子化规则，重点讨论了主符号在曲线内域内出现本征值交叉的情况。

Singular Bohr-Sommerfeld conditions for 1D Toeplitz operators: elliptic case

In this article, we state the Bohr-Sommerfeld conditions around a global minimum of the principal symbol of a self-adjoint semiclassical Toeplitz operator on a compact connected Kähler surface, using an argument of normal form which is obtained thanks to Fourier integral operators. These conditions give an asymptotic expansion of the eigenvalues of the operator in a neighbourhood of fixed size of the singularity. We also recover the usual Bohr-Sommerfeld conditions away from the critical point. We end by investigating an example on the two-dimensional torus.

Bohr-Sommerfeld conditions for Integrable Systems with critical manifolds of focus-focus type

We present a detailed study, in the semi-classical regime $h \to 0$, of microlocal properties of systems of two commuting h-PDO s $P_1(h)$, $P_2(h)$ such that the joint principal symbol $p=(p_1,p_2)$ has a special kind of singularity called a "focus-focus" singularity. Typical examples include the quantum spherical pendulum or the quantum Champagne bottle. In the spirit of Colin de Verdière and Parisse, we show that such systems have a universal behavior described by singular quantization conditions of Bohr-Sommerfeld type. These conditions are used to give a precise description of the joint s…

应变莫尔晶格中的半经典量子化条件

Semiclassical quantization conditions in strained moiré lattices

本文将势阱处标量符号的Bohr-Sommerfeld规则推广至特征值在势阱底部可能合并的矩阵值符号，并应用于研究应变二维蜂窝晶格等莫尔异质结构中近似平带的存在性。