
Suppose \(e^{-Ht}\) is a symmetric Markov semigroup on \(L^ 2(X)\), that is, a positivity preserving self-adjoint contraction semigroup satisfying \(e^{-Ht}1=1\) (where X is a probability space). Suppose too that \(Sp(H)\subseteq \{0,\epsilon \}\cup [1,\infty)\) for some very small \(\epsilon >0\), so that the ground state eigenvalue 0 is almost but not quite degenerate. Such a ''metastable'' semigroup provides an abstract model of some of the properties of a double well Schrödinger operator. It has been previously shown by E. B. Davies that X divides into two metastable regions, each of which is almost invariant under the action of \(e^{-Ht}\). Estimates were obtained relating the leakage of one metastable region into the other to the eigenvalue gap \(\epsilon\). In this paper, the estimates of Davies are refined; estimates for the metastable region leakage are obtained that are asymptotically exact as \(\epsilon\) \(\to 0\), in contrast to the upper and lower bounds of Davies which differed by a factor of six. Furthermore, an optimal estimate is obtained for the first higher order correction term.
Linear symmetric and selfadjoint operators (unbounded), double well Schrödinger operator, Dirichlet forms, eigenvalue estimates, ground state eigenvalue, metastable region leakage, Estimates of eigenvalues in context of PDEs, symmetric Markov semigroup, positivity preserving self-adjoint contraction semigroup, Markov semigroups and applications to diffusion processes
Linear symmetric and selfadjoint operators (unbounded), double well Schrödinger operator, Dirichlet forms, eigenvalue estimates, ground state eigenvalue, metastable region leakage, Estimates of eigenvalues in context of PDEs, symmetric Markov semigroup, positivity preserving self-adjoint contraction semigroup, Markov semigroups and applications to diffusion processes
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