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

V. A. Vinokurov ; V. A. Sadovnichii

Izvestiya. Mathematics , Tome 64 (2000) no. 4, pp. 695-754

Résumé

For the Sturm–Liouville boundary-value problem on a segment we construct asymptotics for $s_n=\sqrt{\lambda_n}$, where $\lambda_n$ are the eigenvalues, and for the normalized eigenfunctions $y_n(x)$ of the form $$ s_n=s_{n,m}(q)+\psi_{n,m}, \qquad y_n(x)=y_{n,m}(q,x)+\Delta y_{n,m}(x) $$ for any $m=0,1,2,\dots$, where $s_{n,m}(q)$ and $y_{n,m}(q,x)$ are expressed explicitly in terms of the potential $q(x)$. Under the assumption that $q(x)$ is a real summable function, the terms $\psi_{n,m}$ and $\Delta y_{n,m}(x)$ are $O\biggl(\dfrac1{n^{m+1}}\biggr)$ as $n\to\infty$.