Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 505

The $Spin^C$ Dirac Operator on High Tensor Powers of a Line Bundle

X. Ma, G. Marinescu

Let $(X,omega)$ be a compact symplectic manifold and let $L$ be a hermitian line bundle over $X$ endowed with a hermitian connection $ abla^L$ with curvature $R^L=-2piimathomega$. Let $E$ be a hermitian vector bundle $E$ on $X$. We study the asymptotic of the spectrum of the square $D^2_k$ of the $ ext{spin}^c$ Dirac operator $D_k$ on $L^kotimes E$. We show that the spectrum of $D^2_k$ is contained in the set ${0}cup(2klambda-C,+infty)$, for some positive constants $lambda>0$, $C>0$. Let us consider the Schr"odinger operator $Delta^#_k=Delta_k-k au$, where $Delta_k$ is the metric Laplacian and $ au(x)=sum_j R^L (w_j,overline{w}_j) >0$, where ${w_j}_{j=1}^n$ is an orthonormal basis of $T_x^{(1,0)}X$. As application, we get a simple proof of a result of Guillemin--Uribe, saying that the spectrum of $Delta^#_k$ is contained in the union $(-a,a)cup(2klambda-b,+infty)$, where $a$ and $b$ are positive constants independent of $k$. The original proof used the analysis of Toeplitz operators of Boutet de Monvel and Guillemin. Let us point out that the decomposition of the spectrum of $Delta^#$ allows Borthwick and Uribe to introduce a new quantization scheme. We also include a generalization to the case of covering manifolds.

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