本文研究了当势函数V在无穷远处不趋于无穷大时，薛定谔算子本征值的渐近性。这种势称为简并势。关键是相空间中的一组相关联的哈密顿量小于一个固定的能量E可能有一个无限大的体积，使外尔公式给出的行为的计数功能必须重新审视。我们回顾在这方面的各种结果，在经典的背景下，以及在半经典的和评论不同的方法。在第3，4节中，我们提出了我们与A Morame（Université de Nantes）的联合工作，涉及退化势V（x）=f（y）g（z），其中g被假定为m个变量的齐次正函数，f是n个变量的光滑严格正函数，最小值在0。当f在无穷远处趋于无穷大时，我们给出了小于固定能量的特征值个数的半经典渐近性质。然后，我们给出了一个尖锐的估计的低本征值，使用玻恩奥本海默近似。通过一种改进的方法，我们也定位了更高的能量。最后，我们将前面的方法应用到一类在正则超曲面上为零的势函数上。
英文摘要：
This paper is devoted to the asymptotics of eigenvalues for a Schrö-dinger operator in the case when the potential V does not tend to infinity at infinity. Such a potential is called degenerate. The point is that the set in the phase space where the associated hamiltonian is smaller than a fixed energy E may have an infinite volume, so that the Weyl formula which gives the behaviour of the counting function has to be revisited. We recall various results in this area, in the classical context as well as in the semi-classical one and comment the different methods. In sections 3, 4 we present our joint works with A Morame, (Université de Nantes),concerning a degenerate potential V(x) =f(y) g(z), where g is assumed to be a homogeneous positive function of m variables, and f is a smooth and strictly positive function of n variables, with a minimum in 0. In the case where f tends to infinity at infinity, we give the semi-classical asymptotic behaviour of the number of eigenvalues less than a fixed energy . Then we give a sharp estimate of the low eigenvalues, using a Born Oppenheimer approximation. With a refined approach we localize also higher energies . Finally we apply the previous methods to a class of potentials which vanish on a regular hypersurface.
A Time-Dependent Born-Oppenheimer Approximation with Exponentially Small Error Estimates
本文构造了分子量子力学的指数精度含时Born-Oppenheimer近似。我们研究的分子系统的电子质量是固定的，其核质量是成正比的$ε^{-4}$，其中$ε$是一个小的扩展参数。通过对渐近展开式的最佳截断，我们构造了含时薛定谔方程的近似解，对于某些C和$γ>0$，该近似解与精确的归一化解一致，直到其范数由$\ds C \exp(-γ/ε^2)$限制的误差为止.
Computational error estimates for Born-Oppenheimer molecular dynamics with nearly crossing potential surfaces
Weierstrass Institute for Applied Analysis and Stochastics（魏尔斯特拉斯应用分析与随机性研究所）；University of Oslo（奥斯陆大学）；Kungl. Tekniska Högskolan（皇家理工学院）；University of Delaware（特拉华大学）
