Eigenvalue Gap Theorems for a Class of Nonsymmetric Elliptic Operators on Convex Domains

Abstract

In the remarkable paper [AC] Andrews and Clutterbuck solve the “gap conjecture”, that is, they show that the difference between the first and second eigenvalues of the laplacian with convex potential on a convex domain in euclidean space is at least 3π 2 D2 . Here D is the diameter of the domain. Somewhat later, Lei Ni [N1] reformulated and expanded some of the techniques introduced in [AC]. Taken together, these papers suggest a general approach to estimating the eigenvalue gap of a large class of linear second-order elliptic operators on convex domains. In this paper we illustrate how this approach may work by estimating the eigenvalue gap of a class of nonsymmetric linear elliptic operators. Let Ω be a strictly convex open domain in euclidean space with smooth boundary. The operators L we consider have the form:

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