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Mat. Zametki, 2002, Volume 72, Issue 2, Pages 292–302 (Mi mz423)  

This article is cited in 20 scientific papers (total in 20 papers)

On the Similarity of Some Differential Operators to Self-Adjoint Ones

M. M. Faddeev, R. G. Shterenberg

Saint-Petersburg State University

Abstract: The paper is devoted to the study of the similarity to self-adjoint operators of operators of the form $L=-\frac {\operatorname {sign}x}{|x|^\alpha p(x)} \frac {d^2}{dx^2}$, $\alpha >-1$, in the space $L_2(\mathbb R)$ with weight $|x|^\alpha p(x)$. As is well known, the answer to this problem in the case $p(x)\equiv 1$ is positive; it was obtained by using delicate methods of the theory of Hilbert spaces with indefinite metric. The use of a general similarity criterion in combination with methods of perturbation theory for differential operators allows us to generalize this result to a much wider class of weight functions $p(x)$.

DOI: https://doi.org/10.4213/mzm423

Full text: PDF file (227 kB)
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English version:
Mathematical Notes, 2002, 72:2, 261–270

Bibliographic databases:

UDC: 517.948
Received: 04.08.2001

Citation: M. M. Faddeev, R. G. Shterenberg, “On the Similarity of Some Differential Operators to Self-Adjoint Ones”, Mat. Zametki, 72:2 (2002), 292–302; Math. Notes, 72:2 (2002), 261–270

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. S. Kostenko, “Similarity of Indefinite Sturm–Liouville Operators with Singular Potential to a Self-Adjoint Operator”, Math. Notes, 78:1 (2005), 134–139  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. Albeverio S, Kuzhel S, “One-dimensional Schrodinger operators with rho-symmetric zero-range potentials”, Journal of Physics A-Mathematical and General, 38:22 (2005), 4975–4988  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    3. A. S. Kostenko, “Similarity of Some $J$-Nonnegative Operators to Self-Adjoint Operators”, Math. Notes, 80:1 (2006), 131–135  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. Kostenko A.S., “Spectral Analysis of Some Indefinite Sturm-Liouville Operators”, Operator Theory 20, Proceedings, eds. Davidson K., Gaspar D., Stratila S., Timotin D., Vasilescu F., Theta Foundation, 2006, 131–141  mathscinet  zmath  isi
    5. Karabash I., Kostenko A., “Spectral Analysis of Differential Operators with Indefinite Weights and a Local Point Interaction”, Operator Theory in Inner Product Spaces, Operator Theory : Advances and Applications, 175, eds. Forster KH., Jones P., Langer H., Trunk C., Birkhauser Verlag Ag, 2007, 169–191  crossref  mathscinet  zmath  isi
    6. Karabash, I, “Indefinite Sturm-Liouville operators with the singular critical point zero”, Proceedings of the Royal Society of Edinburgh Section A-Mathematics, 138 (2008), 801  crossref  mathscinet  zmath  isi  scopus
    7. Karabash, I, “Spectral properties of singular Sturm-Liouville operators with indefinite weight sgn x”, Proceedings of the Royal Society of Edinburgh Section A-Mathematics, 139 (2009), 483  crossref  mathscinet  zmath  isi  scopus  scopus
    8. Karabash, IM, “The similarity problem for J-nonnegative Sturm-Liouville operators”, Journal of Differential Equations, 246:3 (2009), 964  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    9. Karabash I.M., “Abstract Kinetic Equations with Positive Collision Operators”, Spectral Theory in Inner Product Spaces and Applications, Operator Theory Advances and Applications, 188, eds. Behrndt J., Forster KH., Langer H., Trunk C., Birkhauser Verlag Ag, 2009, 175–195  mathscinet  zmath  isi
    10. Markov V.G., “Nekotorye svoistva neznakoopredelennykh operatorov shturma-liuvillya”, Matematicheskie zametki YaGU, 19:1 (2012), 44–59  mathscinet  zmath  elib
    11. Kostenko A., “The Similarity Problem for Indefinite Sturm-Liouville Operators and the Help Inequality”, Adv. Math., 246 (2013), 368–413  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    12. Krejcirik D., Siegl P., Zelezny J., “On the Similarity of Sturm-Liouville Operators with Non-Hermitian Boundary Conditions to Self-Adjoint and Normal Operators”, Complex Anal. Oper. Theory, 8:1 (2014), 255–281  crossref  mathscinet  zmath  isi  scopus  scopus
    13. Pyatkov S.G., “Existence of Maximal Semidefinite Invariant Subspaces and Semigroup Properties of Some Classes of Ordinary Differential Operators”, Oper. Matrices, 8:1 (2014), 237–254  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    14. G. M. Gubreev, A. A. Tarasenko, “On the Similarity to Self-Adjoint Operators”, Funct. Anal. Appl., 48:4 (2014), 286–290  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    15. Gil M., “A Bound For Similarity Condition Numbers of Unbounded Operators With Hilbert-Schmidt Hermitian Components”, J. Aust. Math. Soc., 97:3 (2014), 331–342  crossref  mathscinet  zmath  isi  scopus  scopus
    16. Gil' Michael, “on Condition Numbers of Spectral Operators in a Hilbert Space”, Anal. Math. Phys., 5:4 (2015), 363–372  crossref  mathscinet  zmath  isi  scopus  scopus
    17. Gil' Michael, “An inequality for similarity condition numbers of unbounded operators with Schatten - von Neumann Hermitian components”, Filomat, 30:13 (2016), 3415–3425  crossref  mathscinet  zmath  isi  elib  scopus
    18. Dritschel M.A., Estevez D., Yakubovich D., “Resolvent Criteria For Similarity to a Normal Operator With Spectrum on a Curve”, J. Math. Anal. Appl., 463:1 (2018), 345–364  crossref  mathscinet  zmath  isi  scopus  scopus
    19. Gil' Michael, “On Similarity of Unbounded Perturbations of Selfadjoint Operators”, Methods Funct. Anal. Topol., 24:1 (2018), 27–33  mathscinet  isi
    20. Gil M., “Similarity of Operators on Tensor Products of Spaces and Matrix Differential Operators”, J. Aust. Math. Soc., 106:1 (2019), 19–30  crossref  mathscinet  zmath  isi  scopus
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