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Zh. Vychisl. Mat. Mat. Fiz., 2006, Volume 46, Number 4, Pages 683–699 (Mi zvmmf488)  

This article is cited in 13 scientific papers (total in 13 papers)

Set-valued mappings specified by regularization of the Schrödinger equation with degeneration

V. Zh. Sakbaev

Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, Moscow oblast, 141700, Russia

Abstract: The Cauchy problem for the Schrödinger equation with an operator degenerating on a half-line and a family of regularized Cauchy problems with uniformly elliptic operators, whose solutions approximate the solution to the degenerate problem, are considered. A set-valued mapping is investigated that takes a bounded operator to a set of partial limits of values of its quadratic form on solutions of the regularized problems when the regularization parameter tends to zero. The dynamics of quantum states are determined by applying an averaging procedure to the set-valued mapping.

Key words: degenerating operator, quantum state, set-valued mapping, finitely additive measure.

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English version:
Computational Mathematics and Mathematical Physics, 2006, 46:4, 651–665

Bibliographic databases:

UDC: 519.634
Received: 07.12.2004
Revised: 20.02.2005

Citation: V. Zh. Sakbaev, “Set-valued mappings specified by regularization of the Schrödinger equation with degeneration”, Zh. Vychisl. Mat. Mat. Fiz., 46:4 (2006), 683–699; Comput. Math. Math. Phys., 46:4 (2006), 651–665

Citation in format AMSBIB
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\paper Set-valued mappings specified by regularization of the Schr\"odinger equation with degeneration
\jour Zh. Vychisl. Mat. Mat. Fiz.
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\pages 683--699
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\transl
\jour Comput. Math. Math. Phys.
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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:
    1. V. Zh. Sakbaev, “On the Cauchy problem for the Schrödinger equation degenerating outside a segment: properties of solutions and spectral aspects of the regularization”, Journal of Mathematical Sciences, 153:5 (2008), 562–590  mathnet  crossref  mathscinet  zmath
    2. Sakbaev V.Zh., “Stochastic properties of degenerated quantum systems”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 13:1 (2010), 65–85  crossref  mathscinet  zmath  isi  elib  scopus
    3. V. Zh. Sakbaev, “Averaging of quantum dynamical semigroups”, Theoret. and Math. Phys., 164:3 (2010), 1215–1221  mathnet  crossref  crossref  adsnasa  isi
    4. V. Zh. Sakbaev, “O dinamike mnozhestva sostoyanii kvantovoi sistemy s vyrozhdennym gamiltonianom”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 2(23) (2011), 200–220  mathnet  crossref
    5. V. Z. Sakbaev, “The set of quantum states and its averaged dynamic transformations”, Russian Math. (Iz. VUZ), 55:10 (2011), 41–50  mathnet  crossref  mathscinet  elib
    6. V. Zh. Sakbaev, “Cauchy problem for degenerating linear differential equations and averaging of approximating regularizations”, Journal of Mathematical Sciences, 213:3 (2016), 287–459  mathnet  crossref  mathscinet
    7. Dzh. O. Ogun, Yu. N. Orlov, V. Zh. Sakbaev, “O preobrazovanii prostranstva nachalnykh dannykh dlya zadachi Koshi s osobennostyami resheniya tipa vzryva”, Preprinty IPM im. M. V. Keldysha, 2012, 087, 31 pp.  mathnet
    8. V. Sakbaev, “On dynamical properties of a one-parameter family of transformations arising in averaging of semigroups”, Journal of Mathematical Sciences, 202:6 (2014), 869–886  mathnet  crossref
    9. M. Kh. Numan Elsheikh, D. O. Ogun, Yu. N. Orlov, R. V. Pleshakov, V. Zh. Sakbaev, “Usrednenie sluchainykh polugrupp i neodnoznachnost kvantovaniya gamiltonovykh sistem”, Preprinty IPM im. M. V. Keldysha, 2014, 019, 28 pp.  mathnet
    10. Yu. N. Orlov, V. Zh. Sakbaev, O. G. Smolyanov, “Feynman formulas as a method of averaging random Hamiltonians”, Proc. Steklov Inst. Math., 285 (2014), 222–232  mathnet  crossref  crossref  isi  elib  elib
    11. L. A. Borisov, Yu. N. Orlov, V. Zh. Sakbaev, “Formuly Feinmana dlya usredneniya polugrupp, porozhdaemykh operatorami tipa Shredingera”, Preprinty IPM im. M. V. Keldysha, 2015, 057, 23 pp.  mathnet
    12. L. A. Borisov, Yu. N. Orlov, V. Zh. Sakbaev, “Ekvivalentnost po Chernovu primenitelno k uravneniyam evolyutsii matritsy plotnosti i funktsii Vignera dlya lineinogo kvantovaniya”, Preprinty IPM im. M. V. Keldysha, 2015, 066, 28 pp.  mathnet
    13. Borisov L.A., Orlov Yu.N., Sakbaev V.Zh., “Feynman Averaging of Semigroups Generated By Schrodinger Operators”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 21:2 (2018), 1850010  crossref  mathscinet  zmath  isi  scopus
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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