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

We consider the perturbed Stark operator H q φ = - φ ′ ′ + x φ + q (x) φ , φ (0) = 0 , in L 2 (R +) , where q is a real function that belongs to A r = q ∈ A r ∩ AC [ 0 , ∞) : q ′ ∈ A r , where A r = L R 2 (R + , (1 + x) r d x) and r > 1 is arbitrary but fixed. Let λ n (q) n = 1 ∞ and κ n (q) n = 1 ∞ be 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) = n - 1 / 3 log 1 / 2 n if r ∈ (1 , 2) and ω r (n) = n - 1 / 3 if r ∈ [ 2 , ∞) . We prove that λ n (q) = - a n + π (- a n) - 1 / 2 ∫ 0 ∞ Ai 2 (x + a n) q (x) d x + O (n - 1 / 3 ω r 2 (n)) and κ n (q) = - 2 π (- a n) - 1 / 2 ∫ 0 ∞ Ai (x + a n) Ai ′ (x + a n) q (x) d x + O (ω r 3 (n)) , uniformly on bounded subsets of A r . In order to obtain these asymptotic formulas, we first show that λ n : A r → R and κ n : A r → R are real analytic maps. [ABSTRACT FROM AUTHOR]