On the gauge for the Neumann problem in the half space

Abstract

We consider the gauge function G for the Neumann problem for 1/2Δ⁺q in the half space D = {(α, x) ∈ Rd : α > 0}, where q is independent of α and is periodic in x. It is shown that if G ≠ ∞, then G is a bounded continuous function on Cl(D). If $H(x) = \int_0^{\infty }G(\alpha ,x)d\alpha ≠\∞ $8, it is shown that the corresponding Feynman-Kac semi-group decays exponentially.