Sturm-liouville operators on the whole line, with the same discrete spectrum

B. M. Levitan

Sbornik. Mathematics, Tome 60 (1988) no. 1, pp. 77-106

Résumé

It is proved that all differential operators of the form \begin{equation} -y''+q(x) y=\lambda y \qquad (-\infty\infty) \label{1} \end{equation} whose spectrum $\{\lambda_n\}^\infty_{n=0}$ coincides with the spectrum of the linear oscillator \begin{equation} -y''+(x^2-1)y=\lambda y \qquad (-\infty\infty), \label{2} \end{equation} i.e. $\lambda_n=2n$, $n=0,1,2,\dots$, and whose potentials $q(x)$ are sufficiently smooth and differ sufficiently little from the potential $(x^2-1)$ may be obtained by the well-known method of the theory of the inverse Sturm–Liouville problem. This result was obtained earlier by McKean and Trubowitz (Comm. in Math., 1982, v. 82, p. 471–495). This paper gives another proof of this theorem, based on the following completeness theorem, which is interesting in itself. Denote by $\{e_n(x)\}^\infty_{n=0}$ the eigenfunctions of equation (1) and by $\{e_n^0(x)\}^\infty_{n=0}$ the eigenfunctions of equation (2). The linear span of the set of functions $$ \{e_n(x)e_n^0(x)\}^\infty_{n=0}\cup\{[e_n(x)e_n^0(x)]'\}^\infty_{n=0} $$ is dense in the space $L^2(-\infty,\infty)$. Bibliography: 8 titles.