PROOF OF THE FUNDAMENTAL GAP CONJECTURE

摘要

We consider Schrodinger operators of the form -Δ + V with Dirichlet boundary conditions on a compact convex domain Ω in R~n. The diameter of Ω is given by D = sup_(x,y)∈ Ω||y-x||. We assume that the potential V is semiconvex (i.e., V+c||x||~2 is convex for some c). Such an operator has an increasing sequence of eigenvalues λ_0 < λ_1 <λ_2≤... and corresponding eigenfunctions {φ_i}_i≥0 which vanish on Ω and satisfy the equation (1) Δ_i — Vφ_i λ_iφ_i = 0. The difference between the first two eigenvalues, — A0, is called the fundamental gap. It is of importance for several reasons: In quantum mechanics it represents the 'excitation energy' required to reach the first excited state from the ground state; it thus determines the stability of the ground state and so is also important in statistical mechanics and quantum field theory. The spectral gap also determines the rate at which positive solutions of the heat equation approach the first eigenspace, and it is through this characterization that we will prove the conjecture of the title.