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

TOLOZA, JULIO H.; URIBE, ALFREDO

Analysis and Mathematical Physics

BIRKHAUSER (SPRINGER NATURE)

Año: 2022 vol. 13

1664-2368

We consider the perturbed Stark operator $H_qarphi = -arphi´´ + xarphi + q(x)arphi$, $arphi(0)=0$,in $L^2(R_+)$, where $q$ is a real function that belongs to$sA_r =left{ qinsA_rcapext{AC}[0,infty) : q´insA_right}$,where $sA_r = L^2_R(R_+,(1+x)^r dx)$ and $r>1$ is arbitrary but fixed. Let $left{lambda_n(q)ight}_{n=1}^infty$ and$left{kappa_n(q)ight}_{n=1}^ infty$ be the spectrum and associated set of norming constants of $H_q$. Let ${a_n}_{n=1}^infty$ be the zeros of the Airy function of the first kind, and let $omega_r:NoR$ be defined by the rule$omega_r(n) = n^{-1/3}log^{1/2}n$ if $rin(1,2)$ and $omega_r(n) = n^{-1/3}$ if $rin[2,infty)$. We prove that$lambda_n(q)    = -a_n + pi (-a_n)^{-1/2}int_0^infty ai^2(x+a_n)q(x)dx         + O(n^{-1/3}omega_r^2(n))$and$kappa_n(q)    = - 2pi (-a_n)^{-1/2}int_0^infty ai(x+a_n)ai´(x+a_n)q(x)dx        + O(omega_r^3(n))$,uniformly on bounded subsets of $sA_r$. In order to obtain these asymptotic formulas, we first show that $lambda_n:sA_roR$ and $kappa_n:sA_roR$ are real analytic maps.