Asymptotics of any order for the eigenvalues and eigenfunctions of the Sturm-Liouville boundary-value problem on a segment with a summable potential

For the Sturm-Liouville boundary-value problem on a segment we construct asymptotics for sn=√λn, where λn are the eigenvalues, and for the normalized eigenfunctions yn(x) of the form sn=sn,m(q)+ψn,m, yn(x)=yn,m(q,x)+Δyn,m(x) for any m=0,1,2,..., where sn,m(q) and yn,m(q,x) are expressed explicitly in terms of the potential q(x). Under the assumption that q(x) is a real summable function, the terms ψn,m and Δyn,m are O(1/nm+1) as n→∞