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Sturm-liouville operators on the whole line, with the same discrete spectrum
Sbornik. Mathematics, Tome 60 (1988) no. 1, pp. 77-106 Cet article a éte moissonné depuis la source Math-Net.Ru

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It is proved that all differential operators of the form \begin{equation} -y''+q(x) y=\lambda y \qquad (-\infty<x<\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<x<\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. 
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     title = {Sturm-liouville operators on the whole line, with the same discrete spectrum},
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B. M. Levitan. Sturm-liouville operators on the whole line, with the same discrete spectrum. Sbornik. Mathematics, Tome 60 (1988) no. 1, pp. 77-106. http://geodesic.mathdoc.fr/item/SM_1988_60_1_a5/

[1] McKean H. P., Trubowitz E., “The Spectral Class of the Quantum-Mechanical Harmonic Oscillator”, Comm. in Math., 82 (1982), 471–495 | DOI | MR | Zbl

[2] Segë G., Ortogonalnye mnogochleny, Fizmatgiz, M., 1962

[3] Naimark M. A., Lineinye differentsialnye operatory, Nauka, M., 1969 | MR

[4] Titchmarsh E. Ch., Razlozhenie po sobstvennym funktsiyam, T. 1, IL, M., 1959.

[5] Levitan B. M., Sargsyan I. S., Vvedenie v spektralnuyu teoriyu, Nauka, M., 1970 | MR | Zbl

[6] Levitan B. M., Obratnye zadachi Shturma–Liuvillya, Nauka, M., 1984 | MR

[7] Titchmarsh E. C., “Eigenfunction expansions associated with partial differential equations. III”, Proc. London Math. Soc. (3), 3 (1953), 153–169 | DOI | MR | Zbl

[8] Levitan B. M., “Ob asimptoticheskom povedenii funktsii Grina i razlozhenii po sobstvennym funktsiyam uravneniya Shredingera”, Matem. sb., 41(83) (1957), 439–458 | MR | Zbl