Birkhoff Normal Forms in Semi-Classical Inverse Problems

The semi-classical spectrum and the Birkhoff normal form

The purposes of this note are: 1) to propose a direct and "elementary" proof of the main result proved by Guillemin-Paul-Uribe [GPU], namely that the semi-classical spectrum near a global minimum of the classical Hamiltonian determines the whole semi-classical Birkhoff normal form (denoted the BNF) in the non-resonant case. 2) to present in the completely resonant case a similar problem.

A semiclassical Birkhoff normal form for symplectic magnetic wells

In this paper we construct a Birkhoff normal form for a semiclassical magnetic Schr{ö}dinger operator with non-degenerate magnetic field, and discrete magnetic well, defined on an even dimensional riemannian manifold M. We use this normal form to get an expansion of the first eigenvalues in powers of h^{1/2}, and semiclassical Weyl asymptotics for this operator.

近共振 Schrödinger 算子的半经典 Birkhoff-Gustavson 范式与谱渐近

Semiclassical Birkhoff-Gustavson normal forms and spectral asymptotics for nearly resonant Schrödinger operators

本文引入半经典 Schrödinger 算子的近共振概念，结合 Birkhoff-Gustavson 范式扩展提出精确且易实现的离散谱逼近公式，并应用于近 Fermi 1:2 共振情形进行验证。

Spectral asymptotics via the semiclassical Birkhoff normal form

This article gives a simple treatment of the quantum Birkhoff normal form for semiclassical pseudo-differential operators with smooth coefficients. The normal form is applied to describe the discrete spectrum in a generalised non-degenerate potential well, yielding uniform estimates in the energy $E$. This permits a detailed study of the spectrum in various asymptotic regions of the parameters $(E,\h)$, and gives improvements and new proofs for many of the results in the field. In the completely resonant case we show that the pseudo-differential operator can be reduced to a Toeplitz operator o…

Birkhoff normal forms for Fourier integral operators II

In this work we construct logarithms and Birkhoff normal forms for elliptic Fourier integral operators in the semi-classical limit under more general assumptions than in aprevious work by the first author. The methods are similar but slightly different.

常秩磁场的半经典Birkhoff正规形

A semiclassical Birkhoff normal form for constant-rank magnetic fields

针对带常秩非零磁场的半经典磁拉普拉斯算子，在磁场有唯一非退化势阱的假设下，构造三个逐次Birkhoff正规形描述半经典极限下的谱，得到阈值以下本征值的$\u210f^{1/2}$幂次展开。