Dirichlet operators with convex potentials on the half-line

Abstract

We consider the Dirichlet operator H _t =−d ²/dx ²+q(x) on L ²([t,∞)), where q is a convex potential with q(x)→∞ as x→∞. We show that the eigenvalue gap Γ(t) of H _t is monotone increasing as t increases from −∞ to ∞. We also show that Γ(t) is strictly increasing if q is not linear at infinity. An asymptotic estimate of Γ(t) for quadratic potentials is obtained.