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This article is cited in 16 scientific papers (total in 16 papers)
Sturm-liouville operators on the whole line, with the same discrete spectrum
B. M. Levitan
Abstract:
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. $\lambda_n=2n$, $n=0,1,2,…$, 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 $L^2(-\infty,\infty)$.
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English version:
Mathematics of the USSR-Sbornik, 1988, 60:1, 77–106
Bibliographic databases:
UDC:
517.95
MSC: Primary 34B25; Secondary 34B27, 34B30 Received: 28.05.1985
Citation:
B. M. Levitan, “Sturm-liouville operators on the whole line, with the same discrete spectrum”, Mat. Sb. (N.S.), 132(174):1 (1987), 73–103; Math. USSR-Sb., 60:1 (1988), 77–106
Citation in format AMSBIB
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\by B.~M.~Levitan
\paper Sturm-liouville operators on the whole line, with the same discrete spectrum
\jour Mat. Sb. (N.S.)
\yr 1987
\vol 132(174)
\issue 1
\pages 73--103
\mathnet{http://mi.mathnet.ru/msb1716}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=883914}
\zmath{https://zbmath.org/?q=an:0661.34017|0625.34021}
\transl
\jour Math. USSR-Sb.
\yr 1988
\vol 60
\issue 1
\pages 77--106
\crossref{https://doi.org/10.1070/SM1988v060n01ABEH003157}
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