Semiclassical spectral analysis of Toeplitz operators on symplectic manifolds: the case of discrete wells

Abstract

We consider Toeplitz operators associated with the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle on a compact symplectic manifold. We study the asymptotic behavior, in the semiclassical limit, of low-lying eigenvalues and the corresponding eigenfunctions of a self-adjoint Toeplitz operator under assumption that its principal symbol has a non-degenerate minimum with discrete wells. As an application, we prove upper bounds for low-lying eigenvalues of the Bochner-Laplacian in the semiclassical limit.

36 pages, v3: final version

References in corpus (3)

• Toeplitz operators on symplectic manifolds
• Donaldson's $Q$-operators for symplectic manifolds
• Symbolic calculus for Toeplitz operators with half-forms

Cited by in corpus (3)

• A semiclassical Birkhoff normal form for symplectic magnetic wells
• A semiclassical Birkhoff normal form for constant-rank magnetic fields
• Semiclassical eigenvalue asymptotics for the Bochner Laplacian of a positive line bundle on a symplectic manifold