A Lower Bound on the First Spectral Gap of Schrodinger Operators with Kato Class Measures

摘要

We study Schrodinger operators on R-n formally given by H-mu = -Delta-mu, where mu is a positive, compactly supported measure from the Kato class. Under the assumption that a certain condition on the mu-volume of balls is satisfied and that H-mu has at least two eigenvalues below the essential spectrum sigma(ess)(H-mu) = [0,infinity), we derive a lower bound on the first spectral gap of H-mu. The assumption on the mu-volume of balls is in particular satisfied if mu is of the form mu = a sigma(M), where M is a compact (n-1)-dimensional Lipschitz submanifold of R-n, sigma(M) the surface measure on M, and 0 <= a is an element of L-infinity(M).