Abstract
In the semiclassical limit hbar rightarrow 0, we analyze a class of self-adjoint Schrödinger operators H_hbar = hbar ^2 L + hbar W + Vcdot {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.
Highlights
We study the low lying spectrum of a Schrödinger operator H on a vector bundle E over a smooth oriented Riemannian manifold M
The endomorphism W is non-vanishing, which is the reason for us to include this term in our considerations. This operator has been considered in detail in [16], giving much mathematical detail to the original paper [31]; see [13], which derives full asymptotic expansions of all low-lying eigenvalues using an inductive approach which builds on the results of [2,3,9] for generators of reversible diffusion operators, using a potential theoretic approach based on estimating capacities
To establish the lower bound, one uses the mini-max formula again and derives a lower bound in terms of a suitable symmetric finite rank operator (constructed from the restrictions of all localized operators Hm j, to the spectral subspace of enery below e ∈, which implies (2.7). These arguments belong to abstract spectral theory and do not depend on the geometry encoded in H and E
Summary
W := Lgrad φ +L∗grad φ is C ∞(M) linear and an endomorphism field as described above In this particular case, the endomorphism W is non-vanishing, which is the reason for us to include this (somewhat unusual) term in our considerations. The endomorphism W is non-vanishing, which is the reason for us to include this (somewhat unusual) term in our considerations This operator has been considered in detail in [16], giving much mathematical detail to the original paper [31]; see [13], which derives full asymptotic expansions of all low-lying eigenvalues using an inductive approach which builds on the results of [2,3,9] for generators of reversible diffusion operators, using a potential theoretic approach based on estimating capacities.
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