论文标题
Bochner-Schrödinger操作员在有界几何形状的象征流形上的半经典光谱分析
Semiclassical spectral analysis of the Bochner-Schrödinger operator on symplectic manifolds of bounded geometry
论文作者
论文摘要
我们研究了Bochner-Schrödinger运算符$ h_ {p} = \ frac1pδ^{l^p \ otimes e}+v $在高张量的高张式幂上，在有界几何的符号歧管上的正线束$ l $。首先，我们根据模型操作员的光谱对其光谱进行粗略的渐近描述。这使我们能够在某些条件下在线束的曲率上证明频谱中的间隙存在。然后，我们考虑了这种操作员的光谱投影，该操作员对应于一个间隔，该间隔不在其核的光谱和研究渐近行为中。首先，我们建立了非对角线指数估计。然后，我们在对角线的固定邻居中陈述了一个完整的渐近扩张。
We study the Bochner-Schrödinger operator $H_{p}=\frac 1pΔ^{L^p\otimes E}+V$ on high tensor powers of a positive line bundle $L$ on a symplectic manifold of bounded geometry. First, we give a rough asymptotic description of its spectrum in terms of the spectra of the model operators. This allows us to prove the existence of gaps in the spectrum under some conditions on the curvature of the line bundle. Then we consider the spectral projection of such an operator corresponding to an interval whose extreme points are not in the spectrum and study asymptotic behavior of its kernel. First, we establish the off-diagonal exponential estimate. Then we state a complete asymptotic expansion in a fixed neighborhood of the diagonal.
