Spectral asymptotics via the semiclassical Birkhoff normal form
Spectral asymptotics via the semiclassical Birkhoff normal form
复制标题
通过半经典 Birkhoff 范式的谱渐进
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
San Vũ Ngọc
中科院分区：
文献类型：
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作者：
L. Charles;San Vũ Ngọc
本文给出了具有光滑系数的半经典伪微分算子的量子Birkhoff范式的一个简单处理。应用标准形式描述广义非简并势阱中的离散谱，得到能量$E$的统一估计。这允许对参数$(E,h)$的各种渐近区域的谱进行详细的研究，并对该领域的许多结果给出改进和新的证明。在完全共振情况下，我们证明了伪微分算子可以约化为约化辛轨道上的Toeplitz算子。利用这种量子约简，证明了特征值簇精细结构的谱渐近性。在多项式微分算子的情况下，得到了一个组合轨迹公式。
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 on a reduced symplectic orbifold. Using this quantum reduction, new spectral asymptotics concerning the fine structure of eigenvalue clusters are proved. In the case of polynomial differential operators, a combinatorial trace formula is obtained.