Semiclassical spectral analysis of the Bochner-Schrödinger operator on symplectic manifolds of bounded geometry (Q2064164)

Let \((L,h^L)\) be a Hermitian line bundle with bounded geometry on \(X\), a smooth Riemannian manifold of bounded geometry endowed with a Hermitian connection \(\nabla^L\) and \((E,h^E)\) be a Hermitian vector bundle of rank \(k\) with bounded geometry on \(X\) with a Hermitian connection \(\nabla^E\). Let \(p\in\mathbb N\), we denote by \(L_p\) the \(pth\) tensor power of \(L\) and \(\nabla^{L_p\otimes E}\) be the \(C^\infty(X,T^*X\otimes L_p\otimes E)\)-valued Hermitian connection on \(C^\infty(X,L_p\otimes E)\). The induced Bochner Laplacian operator is defined by \(\Delta^{L_p\otimes E}=(\nabla^{L_p\otimes E})^*\nabla^{L_p\otimes E}\) where \((\nabla^{L_p\otimes E})^*\) is the formal adjoint associated to \(\nabla^{L_p\otimes E}\). Let \(V\) be a self-adjoint endomorphism of \(E\). The purpose of the author is to study properties of the operator \(H_p=\frac{1}{p}\Delta^{\nabla^{L_p\otimes E}}+V\) on \(C^\infty(X,L_p\otimes E)\). E.g., he provides an asymptotic description of the spectrum of \(H_p\) (as \(p\) goes to the infinity) in terms of the spectra of the model operators, i.e., some suitable second order differential operators obtained from \(H_p\).