具有对称性的约化Gutzwiller公式：有限群情形
Reduced Gutzwiller formula with symmetry: case of a finite group
摘要：
我们考虑$\mathbb{R}^{2d}$上的经典哈密顿量$H$，它不被有限对称群$G$所改变，其Weyl量子化$\hat{H}$是$L^2(\mathbb{R}^d)$上的自伴算子.若$χ$是$G$的一个不可约特征标，本文研究了由Peter-Weyl分解得到的$\hat{H}\_χ$到$L^2(\mathbb{R}^d)$的对称子空间$L^2\_χ(\mathbb{R}^d)$的谱.我们给出了描述$\hat{H}\_χ$在非临界能量$E\in\mathbb{R}$附近的谱的正则化谱密度的约化半经典渐近性。如果$Σ\_E:=\{H=E \}$是紧的，假设周期轨道在$Σ\_E/G$中是非退化的，我们得到了一个约化的Gutzwiller迹公式，使得约化空间$Σ\_E/G$的周期轨道出现.该方法基于相干态的使用，其传播在M。Combescure和D.罗伯特
英文摘要：
We consider a classical Hamiltonian $H$ on $\mathbb{R}^{2d}$, invariant by a finite group of symmetry $G$, whose Weyl quantization $\hat{H}$ is a selfadjoint operator on $L^2(\mathbb{R}^d)$. If $χ$ is an irreducible character of $G$, we investigate the spectrum of its restriction $\hat{H}\_χ$ to the symmetry subspace $L^2\_χ(\mathbb{R}^d)$ of $L^2(\mathbb{R}^d)$ coming from the decomposition of Peter-Weyl. We give reduced semi-classical asymptotics of a regularised spectral density describing the spectrum of $\hat{H}\_χ$ near a non critical energy $E\in\mathbb{R}$. If $Σ\_E:=\{H=E \}$ is compact, assuming that periodic orbits are non-degenerate in $Σ\_E/G$, we get a reduced Gutzwiller trace formula which makes periodic orbits of the reduced space $Σ\_E/G$ appear. The method is based upon the use of coherent states, whose propagation was given in the work of M. Combescure and D. Robert.
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