Semiclassical spectral estimates for Toeplitz operators
Let \(X\) be a compact Kähler manifold of real dimension \(2n\) and \(L\to X\) a holomorphic Hermitian line bundle such that the curvature of its natural connection is a Kähler form of \(X\). The quantization of \(X\) with Planck's constant \(1/k\) is the space \(H^0(x,L^{\otimes k})\), and the Toeplitz quantization of a classical Hamiltonian function \(H\in C^\infty(X)\) is \(S_k= \pi_k M_H\pi_k\), where \(\pi_k: L^2(X,L^{\otimes k})\to H^0(X, L^{\otimes k})\) is the orthogonal projector and \(M_H\) is the operator of multiplication by \(H\). Let \(P\subset L^*\) be the unit circle bundle in the dual of \(L\). By the condition that the curvature of \(L\) be the Kähler form, \(P\) is strictly pseudoconvex and the contact form \(\alpha\) is the connection form on \(P\). Let \(\partial_\theta\) denote the contact vector field on \(P\), uniquely determined by \(\alpha\) (\(\partial_\theta\) generates a free \(S^1\) \((=\mathbb{R}/2\pi \mathbb{Z})\) action of automorphisms of \(P\)). The operator \(\partial_\theta\) determines the decomposition of the Hardy space of \(P\) into eigenspaces of \(\partial_\theta: {\mathcal H}=\oplus{\mathcal H}_k\). In this decomposition one can naturally make the identification: \({\mathcal H}_k= H^0(X, L^{\otimes k})\). Under this identification \(S_k\) gets identified with the operator on \({\mathcal H}_k\) with eigenvalues and eigenfunctions \((E_i^k, \psi_i^k)\). Let \(\phi\) be Hamiltonian flow of \(H\) on \(X\) and \(\pi:P\to X\) the obvious projection. The first main result (Theorem 1.1) is concerned with the integral kernel of \(\varphi(k(T_k- E))\), where \(T\) is the Toeplitz operator on \({\mathcal H}\), \(T_k= T|{\mathcal H}_k\) and \(\varphi\) is a smooth test function with compactly supported Fourier transform, and the second one (Theorem 1.2) concerns the trace of \(\varphi (k(T_k-E))\). Theorem 1.1. Let \(p_1,p_2\in P\), \(x_i= \pi(p_i)\) and in case \(p_1=p_2\) the point \(x_1= x_2\) is not fixed of \(\phi\). Then: (1) if \(H(x_1)= H(x_2)= E\) and \(x_1,x_2\) lie on the same orbit of \(\phi\), then the sums \((*)\) \(\sum_{i=0}^\infty \varphi(k(T_k- E))\psi_i^k (p_1) \overline{\psi_i^k (p_2)}\) admit an asymptotic expansion as \(k\to\infty\) of the form \[ \sum_r \sum_{j=0}^\infty C_{r,j} \widehat{\varphi}(r) e^{ikt_r} k^{n-{1/2}-y}, \] where \(r\in \mathbb{R}\) and \(\phi_r(x_2)= x_1\). (2) If \(\{x_1:x_2\} \not\subset H^{-1}(E)\) or \(x_1,x_2\) are not connected by any \(\phi\)-trajectory, then \((*)\) decreases rapidly in \(k\).
scientific article; zbMATH DE number 1220358
scientific article