Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

Wave model of the Sturm-Liouville operator on the half-line


Authors: M. I. Belishev and S. A. Simonov
Translated by: the authors
Original publication: Algebra i Analiz, tom 29 (2017), nomer 2.
Journal: St. Petersburg Math. J. 29 (2018), 227-248
MSC (2010): Primary 34B24
DOI: https://doi.org/10.1090/spmj/1491
Published electronically: March 12, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The notion of the wave spectrum of a semibounded symmetric operator was introduced by one of the authors in 2013. The wave spectrum is a topological space determined by the operator in a canonical way. The definition involves a dynamical system associated with the operator: the wave spectrum is constructed from its reachable sets. In the paper, a description is given for the wave spectrum of the operator $ L_0=-\frac {d^2}{dx^2}+q$ that acts in the space $ L_2(0,\infty )$ and has defect indices $ (1,1)$. A functional (wave) model is constructed for the operator $ L_0^*$ in which the elements of the original $ L_2(0,\infty )$ are realized as functions on the wave spectrum. This model turns out to be identical to the original $ L_0^*$. The latter is fundamental in solving inverse problems: the wave model is determined by their data, which allows reconstruction of the original.


References [Enhancements On Off] (What's this?)

  • 1. M. I. Belishev, On a problem of M. Kac on the reconstruction of the shape of a domain from the spectrum of the Dirichlet problem, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 173 (1988), no. Mat. Vopr. Teor. Rasprostr. Voln. 18, 30–41, 180 (Russian, with English summary); English transl., J. Soviet Math. 55 (1991), no. 3, 1663–1672. MR 983352, https://doi.org/10.1007/BF01098204
  • 2. M. I. Belishev and M. N. Demchenko, A dynamical system with a boundary control associated with a semibounded symmetric operator, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 409 (2012), no. Matematicheskie Voprosy Teorii Rasprostraneniya Voln. 42, 17–39, 240–241 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 194 (2013), no. 1, 8–20. MR 3032226
  • 3. Garrett Birkhoff, Lattice theory, Third edition. American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR 0227053
  • 4. M. Š. Birman and M. Z. Solomjak, \cyr Spektral′naya teoriya samosopryazhennykh operatorov v gil′bertovom prostranstve, Leningrad. Univ., Leningrad, 1980 (Russian). MR 609148
  • 5. M. I. Višik, On general boundary problems for elliptic differential equations, Trudy Moskov. Mat. Obšč. 1 (1952), 187–246 (Russian). MR 0051404
  • 6. John L. Kelley, General topology, Springer-Verlag, New York-Berlin, 1975. Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.]; Graduate Texts in Mathematics, No. 27. MR 0370454
  • 7. A. N. Kolmogorov and S. V. Fomin, Elements of the theory of functions and functional analysis. Vol. 1. Metric and normed spaces, Graylock Press, Rochester, N. Y., 1957. Translated from the first Russian edition by Leo F. Boron. MR 0085462
  • 8. A. N. Kočubeĭ, Extensions of symmetric operators and of symmetric binary relations, Mat. Zametki 17 (1975), 41–48 (Russian). MR 0365218
  • 9. M. A. Naĭmark, \cyr Lineĭnye differentsial′nye operatory, Izdat. “Nauka”, Moscow, 1969 (Russian). Second edition, revised and augmented; With an appendix by V. È. Ljance. MR 0353061
  • 10. A. V. Shtraus, Functional models and generalized spectral functions of symmetric operators, Algebra i Analiz 10 (1998), no. 5, 1–76 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 10 (1999), no. 5, 733–784. MR 1660000
  • 11. M. I. Belishev, Boundary control in reconstruction of manifolds and metrics (the BC method), Inverse Problems 13 (1997), no. 5, R1–R45. MR 1474359, https://doi.org/10.1088/0266-5611/13/5/002
  • 12. -, Recent progress in the boundary control method, Inverse Problems 23 (2007), no. 5, R1-R67. MR 2008h:93001
  • 13. -, Nest spectrum of symmetric operator and reconstruction of manifolds, Proc. Workshops Inverse Problems, Data, Math. Stat. and Ecology, LiTH-MAT-R-2011/11-SE, Dep. Math. Linköping Univ., Linköping, 2011, pp. 6-14.
  • 14. M. I. Belishev, A unitary invariant of a semi-bounded operator in reconstruction of manifolds, J. Operator Theory 69 (2013), no. 2, 299–326. MR 3053343, https://doi.org/10.7900/jot.2010oct22.1925
  • 15. M. I. Belishev and M. N. Demchenko, Elements of noncommutative geometry in inverse problems on manifolds, J. Geom. Phys. 78 (2014), 29–47. MR 3170309, https://doi.org/10.1016/j.geomphys.2014.01.008
  • 16. V. A. Derkach and M. M. Malamud, The extension theory of Hermitian operators and the moment problem, J. Math. Sci. 73 (1995), no. 2, 141–242. Analysis. 3. MR 1318517, https://doi.org/10.1007/BF02367240
  • 17. Ju Myung Kim, Compactness in ℬ(𝒳), J. Math. Anal. Appl. 320 (2006), no. 2, 619–631. MR 2225981, https://doi.org/10.1016/j.jmaa.2005.07.024
  • 18. Vladimir Ryzhov, A general boundary value problem and its Weyl function, Opuscula Math. 27 (2007), no. 2, 305–331. MR 2345962

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 34B24

Retrieve articles in all journals with MSC (2010): 34B24


Additional Information

M. I. Belishev
Affiliation: St. Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg 199034, Russia; St. Petersburg Branch, V. A. Steklov Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg, 191023 Russia
Email: belishev@pdmi.ras.ru

S. A. Simonov
Affiliation: St. Petersburg Branch, V. A. Steklov Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia; St. Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg 199034, Russia; St. Petersburg State Technological Institute (Technical University), Moskovsky pr. 26, St. Petersburg 190013, Russia
Email: sergey.a.simonov@gmail.com

DOI: https://doi.org/10.1090/spmj/1491
Keywords: Functional model of a symmetric operator, Green's system, wave spectrum, inverse problem.
Received by editor(s): October 20, 2016
Published electronically: March 12, 2018
Additional Notes: The first author was supported by RFBR (grant no. 14-01-00535) and by Volkswagen Foundation-2016
The second author was supported by RFBR (grants nos. 16-01-00443 and 16-01-00635)
Dedicated: Dedicated to the memory of V. S. Buslaev
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society