The Dirichlet problem for perturbed Stark operators in the half-line

Abstract

We consider the perturbed Stark operator H q ϕ = −ϕ ′′ + xϕ + q(x)ϕ, ϕ(0) = 0, in L 2 (R + ), where q is a real function that belongs tobe the spectrum and associated set of norming constants of H q . Let {a n } n=1 be the zeros of the Airy function of the first kind, and let ω r : N → R be defined by the rule ω r (n, uniformly on bounded subsets of A r . Along the way, we also show that λ n : A r → R and κ n : A r → R are real analytic maps.