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

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

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English version:
Mathematical Notes, 2002, 72:2, 261–270

Bibliographic databases:

UDC: 517.948

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
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
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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
3. A. S. Kostenko, “Similarity of Some $J$-Nonnegative Operators to Self-Adjoint Operators”, Math. Notes, 80:1 (2006), 131–135
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
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
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
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
8. Karabash, IM, “The similarity problem for J-nonnegative Sturm-Liouville operators”, Journal of Differential Equations, 246:3 (2009), 964
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
10. Markov V.G., “Nekotorye svoistva neznakoopredelennykh operatorov shturma-liuvillya”, Matematicheskie zametki YaGU, 19:1 (2012), 44–59
11. Kostenko A., “The Similarity Problem for Indefinite Sturm-Liouville Operators and the Help Inequality”, Adv. Math., 246 (2013), 368–413
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
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
14. G. M. Gubreev, A. A. Tarasenko, “On the Similarity to Self-Adjoint Operators”, Funct. Anal. Appl., 48:4 (2014), 286–290
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
16. Gil' Michael, “on Condition Numbers of Spectral Operators in a Hilbert Space”, Anal. Math. Phys., 5:4 (2015), 363–372
17. Gil' Michael, “An inequality for similarity condition numbers of unbounded operators with Schatten - von Neumann Hermitian components”, Filomat, 30:13 (2016), 3415–3425
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
19. Gil' Michael, “On Similarity of Unbounded Perturbations of Selfadjoint Operators”, Methods Funct. Anal. Topol., 24:1 (2018), 27–33
20. Gil M., “Similarity of Operators on Tensor Products of Spaces and Matrix Differential Operators”, J. Aust. Math. Soc., 106:1 (2019), 19–30
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