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

Abstract

We consider the perturbed Stark operator Hqφ=−φ′′+xφ+q(x)φ , φ(0)=0 , in L2(R+) , where q is a real function that belongs to Ar={q∈Ar∩AC[0,∞):q′∈Ar} , where Ar=L2R(R+,(1+x)rdx) and r>1 is arbitrary but fixed. Let {λn(q)}∞n=1 and {κn(q)}∞n=1 be the spectrum and associated set of norming constants of Hq . Let {an}∞n=1 be the zeros of the Airy function of the first kind, and let ωr:N→R be defined by the rule ωr(n)=n−1/3log1/2n if r∈(1,2) and ωr(n)=n−1/3 if r∈[2,∞) . We prove that λn(q)=−an+π(−an)−1/2∫∞0Ai2(x+an)q(x)dx+O(n−1/3ω2r(n)) and κn(q)=−2π(−an)−1/2∫∞0Ai(x+an)Ai′(x+an)q(x)dx+O(ω3r(n)) , uniformly on bounded subsets of Ar . In order to obtain these asymptotic formulas, we first show that λn:Ar→R and κn:Ar→R are real analytic maps.