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Mathematics of the USSR-Sbornik, 1977, Volume 31, Issue 4, Pages 457–478
DOI: https://doi.org/10.1070/SM1977v031n04ABEH003716
(Mi sm2695)
 

This article is cited in 52 scientific papers (total in 52 papers)

Selfadjoint dilatation of the dissipative Shrödinger operator and its resolution in terms of eigenfunctions

B. S. Pavlov
References:
Abstract: The object of the present work is the imbedding of the spectral theory for the dissipative Schrödinger operator $L$ with absolutely continuous spectrum acting in the Hilbert space $H=L_2(R^3)$ in the spectral theory of a model operator and the proof of the theorem on expansion in terms of eigenfunctions. The imbedding mentioned is achieved by constructing a selfadjoint dilation $\mathscr L$ of the operator $L$. In the so-called incoming spectral representation of this dilation the operator becomes the corresponding model operator. Next, a system of eigenfunctions of the dilation – the “radiating” eigenfunctions – is constructed. From these a canonical system of eigenfunctions for the absolutely continuous spectrum of the operator and its spectral projections are obtained by “orthogonal projection” onto $H$.
Bibliography: 22 titles.
Received: 11.03.1976
Bibliographic databases:
UDC: 517.43
MSC: Primary 35J10, 35P10, 47A20; Secondary 47B44, 35P25
Language: English
Original paper language: Russian
Citation: B. S. Pavlov, “Selfadjoint dilatation of the dissipative Shrödinger operator and its resolution in terms of eigenfunctions”, Math. USSR-Sb., 31:4 (1977), 457–478
Citation in format AMSBIB
\Bibitem{Pav77}
\by B.~S.~Pavlov
\paper Selfadjoint dilatation of the dissipative Shr\"odinger operator and its resolution in terms of eigenfunctions
\jour Math. USSR-Sb.
\yr 1977
\vol 31
\issue 4
\pages 457--478
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\zmath{https://zbmath.org/?q=an:0356.47007|0388.47003}
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Linking options:
  • https://www.mathnet.ru/eng/sm2695
  • https://doi.org/10.1070/SM1977v031n04ABEH003716
  • https://www.mathnet.ru/eng/sm/v144/i4/p511
  • This publication is cited in the following 52 articles:
    1. Vladimir A. Zolotarev, Lecture Notes in Mathematics, 2355, Analytic Methods of Spectral Representations of Non-Selfadjoint (Non-Unitary) Operators, 2025, 455  crossref
    2. Vladimir A. Zolotarev, Lecture Notes in Mathematics, 2355, Analytic Methods of Spectral Representations of Non-Selfadjoint (Non-Unitary) Operators, 2025, 1  crossref
    3. Vladimir A. Zolotarev, Lecture Notes in Mathematics, 2355, Analytic Methods of Spectral Representations of Non-Selfadjoint (Non-Unitary) Operators, 2025, 285  crossref
    4. Kirill D. Cherednichenko, Yulia Yu. Ershova, Sergey N. Naboko, “Functional model for generalised resolvents and its application to time-dispersive media”, Anal.Math.Phys., 14:6 (2024)  crossref
    5. St. Petersburg Math. J., 35:1 (2024), 25–59  mathnet  crossref
    6. St. Petersburg Math. J., 35:1 (2024), 217–232  mathnet  crossref
    7. Sergio Albeverio, Volodymyr Derkach, Mark Malamud, Operator Theory: Advances and Applications, 291, From Complex Analysis to Operator Theory: A Panorama, 2023, 75  crossref
    8. D. V. Tretyakov, Yu. L. Kudryashov, “On a General Approach to Construction of a Self-Adjoint Dilation for a Dissipative Operator”, J Math Sci, 268:6 (2022), 816  crossref
    9. D. V. Tretyakov, Yu. L. Kudryashov, “Ob obschem podkhode k postroeniyu samosopryazhennoi dilatatsii dissipativnogo operatora”, Issledovaniya po lineinym operatoram i teorii funktsii. 49, Zap. nauchn. sem. POMI, 503, POMI, SPb., 2021, 121–136  mathnet
    10. Yu. L. Kudryashov, “Dilatatsii lineinykh operatorov”, Trudy Krymskoi osennei matematicheskoi shkoly-simpoziuma, SMFN, 66, no. 2, Rossiiskii universitet druzhby narodov, M., 2020, 209–220  mathnet  crossref
    11. B. P. Allahverdiev, H. Tuna, “Dissipative Dirac operator with general boundary conditions on time scales”, Ukr. Mat. Zhurn., 72:5 (2020)  crossref
    12. B. P. Allahverdiev, H. Tuna, “Dissipative Dirac Operator with General Boundary Conditions on Time Scales”, Ukr Math J, 72:5 (2020), 671  crossref
    13. Yu. M. Arlinskiǐ, “Compressed Resolvents, Schur Functions, Nevanlinna Families and Their Transformations”, Complex Anal. Oper. Theory, 14:6 (2020)  crossref
    14. Bilender P. Allahverdiev, Hüseyin Tuna, “Dissipative q-Dirac operator with general boundary conditions”, Quaestiones Mathematicae, 41:2 (2018), 239  crossref
    15. Aytekin Ery{\i}lmaz, Hüseyin Tuna, “Spectral analysis of dissipative fractional Sturm–Liouville operators”, Georgian Mathematical Journal, 24:3 (2017), 351  crossref
    16. Royer J., “Mourre's commutators method for a dissipative form perturbation”, J. Operat. Theor., 76:2 (2016), 351–385  crossref  mathscinet  zmath  isi  elib  scopus
    17. Y. Strauss, “A modified Lax-Phillips scattering theory for quantum mechanics”, J. Math. Phys, 56:7 (2015), 073501  crossref  mathscinet  zmath
    18. Wang X.P., Zhu L., “On the Wave Operator for Dissipative Potentials with Small Imaginary Part”, Asymptotic Anal., 86:1 (2014), 49–57  crossref  mathscinet  zmath  isi
    19. Aytekin Eryilmaz, Hüseyin Tuna, “Spectral theory of dissipative q-Sturm-Liouville problems”, Studia Scientiarum Mathematicarum Hungarica, 51:3 (2014), 366  crossref  mathscinet  zmath
    20. Hüseyin Tuna, “On spectral properties of dissipative fourth order boundary-value problem with a spectral parameter in the boundary condition”, Applied Mathematics and Computation, 219:17 (2013), 9377  crossref  mathscinet
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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