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Mat. Sb. (N.S.), 1987, Volume 132(174), Number 1, Pages 73–103 (Mi msb1716)  

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 $L^2(-\infty,\infty)$.
Bibliography: 8 titles.

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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
\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
\jour Math. USSR-Sb.
\yr 1988
\vol 60
\issue 1
\pages 77--106

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    This publication is cited in the following articles:
    1. Skoblin YA., “On a Class of Sturm-Liouville Operators on a Semiaxis with Given Spectrum”, Vestn. Mosk. Univ. Seriya 1 Mat. Mekhanika, 1987, no. 6, 59–62  mathscinet  isi
    2. Gasymov M. Levitan B., “On the Expansion on Products of Special Solutions of 2 Sturm-Liouville Equations”, 310, no. 4, 1990, 781–784  mathscinet  zmath  isi
    3. Mishev Y., “Crum-Krein Transforms and Lambda-Operators for Radial Schrodinger-Equations”, Inverse Probl., 7:3 (1991), 379–398  crossref  mathscinet  zmath  adsnasa  isi
    4. Daskalov V., “On the Inverse Problems for the Regular Dirac Operator”, 45, no. 11, 1992, 15–18  mathscinet  zmath  isi
    5. Daskalov V., “On the Inverse Problems for the Regular Sturm-Liouville Operator”, 45, no. 10, 1992, 17–20  mathscinet  zmath  isi
    6. A. P. Veselov, A. B. Shabat, “Dressing Chains and Spectral Theory of the Schrödinger Operator”, Funct. Anal. Appl., 27:2 (1993), 81–96  mathnet  crossref  mathscinet  zmath  isi
    7. F. Gesztesy, B. Simon, G. Teschl, “Spectral deformations of one-dimensional Schrödinger operators”, J Anal Math, 70:1 (1996), 267  crossref  mathscinet  zmath  isi
    8. Eleonskii V., Korolev V., “Nonlinear Generalization of Fock Approach to the Analysis of Quantum Systems with Pointed Spectrum”, Zhurnal Eksperimentalnoi Teor. Fiz., 110:6 (1996), 1967–1987  mathscinet  isi
    9. Eleonsky V., Korolev V., “Isospectral Deformation of Quantum Potentials and the Liouville Equation”, Phys. Rev. A, 55:4 (1997), 2580–2593  crossref  mathscinet  adsnasa  isi
    10. Korolev V., “Isospectral Problem: Interplay Between Liouville Equations, Darboux Transforms and Mckean-Trubowitz Flows”, J. Phys. A-Math. Gen., 31:46 (1998), 9297–9307  crossref  mathscinet  zmath  adsnasa  isi
    11. Eleonsky V., Korolev V., “Isospectral Problem for Schrodinger Operator: Evolutional Viewpoint”, J. Math. Phys., 40:4 (1999), 1977–1992  crossref  mathscinet  zmath  adsnasa  isi
    12. Dmitri Chelkak, Pavel Kargaev, Evgeni Korotyaev, “Inverse Problem for Harmonic Oscillator Perturbed by Potential, Characterization”, Commun. Math. Phys, 249:1 (2004), 133  crossref  mathscinet  zmath
    13. M. Asorey, J.F. Cariñena, G. Marmo, A. Perelomov, “Isoperiodic classical systems and their quantum counterparts”, Annals of Physics, 322:6 (2007), 1444  crossref  mathscinet  zmath  elib
    14. I.H.. Chan, T.G.. Shepherd, “Balance model for equatorial long waves”, J. Fluid Mech, 725 (2013), 55  crossref  mathscinet  zmath
    15. I. M. Guseinov, A. Kh. Khanmamedov, A. F. Mamedova, “Inverse scattering problem for the Schrödinger equation with an additional quadratic potential on the entire axis”, Theoret. and Math. Phys., 195:1 (2018), 538–547  mathnet  crossref  crossref  adsnasa  isi  elib
    16. G. M. Masmaliev, A. Kh. Khanmamedov, “Transformation Operators for Perturbed Harmonic Oscillators”, Math. Notes, 105:5 (2019), 728–733  mathnet  crossref  crossref  isi  elib
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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