In the semiclassical limit \(\hbar \rightarrow 0\), we analyze a class of self-adjoint Schrödinger operators \(H_\hbar = \hbar ^2 L + \hbar W + V\cdot {\mathrm {id}}_{\mathscr {E}}\) acting on sections of a vector bundle \({\mathscr {E}}\) over an oriented Riemannian manifold M where L is a Laplace type operator, W is an endomorphism field and the potential energy V has non-degenerate minima at a finite number of points \(m^1,\ldots m^r \in M\), called potential wells. Using quasimodes of WKB-type near \(m^j\) for eigenfunctions associated with the low lying eigenvalues of \(H_\hbar \), we analyze the tunneling effect, i.e. the splitting between low lying eigenvalues, which e.g. arises in certain symmetric configurations. Technically, we treat the coupling between different potential wells by an interaction matrix and we consider the case of a single minimal geodesic (with respect to the associated Agmon metric) connecting two potential wells and the case of a submanifold of minimal geodesics of dimension \(\ell + 1\). This dimension \(\ell \) determines the polynomial prefactor for exponentially small eigenvalue splitting.

在半经典极限\（\ hbar \ rightarrow 0 \）中，我们分析了一类自伴Schrödinger算子\（H_ \ hbar = \ hbar ^ 2 L + \ hbar W + V \ cdot {\ mathrm {id}} _ {\ mathscr {E}} \）作用于定向黎曼流形M上矢量束\（{\ mathscr {E}} \\}的截面上，其中L是拉普拉斯型算子，W是内同态场，势能量V在称为势阱的有限数量的点\（m ^ 1，\ ldots m ^ r \ in M \）中具有非退化极小值。将\（m ^ j \）附近的WKB型准模式用于与\（H_ \ hbar \）的低特征值相关的本征函数，我们分析了隧穿效应，即低位特征值之间的分裂，例如在某些对称配置中出现的现象。从技术上讲，我们通过交互矩阵处理不同势阱之间的耦合，并考虑连接两个势阱的单个最小测地线（相对于相关的Agmon度量）的情况以及尺寸为\（的最小测地线的子流形）的情况。\ ell + 1 \）。此维\（\ ell \）确定用于指数小特征值分裂的多项式前置因子。