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CMFD, 2012, Volume 43, Pages 3–172 (Mi cmfd207)  

This article is cited in 15 scientific papers (total in 15 papers)

Cauchy problem for degenerating linear differential equations and averaging of approximating regularizations

V. Zh. Sakbaev

Moscow Institute of Physics and Engineering, Moscow, Russia

Abstract: In this work, we consider the Cauchy problem for the Schrödinger equation. The generating operator $\mathbf L$ for this equation is a symmetric linear differential operator in the Hilbert space $H=L_2(\mathbb R^d)$, $d\in\mathbb N$, degenerated on some subset of the coordinate space. To study the Cauchy problem when conditions of existence of the solution are violated, we extend the notion of a solution and change the statement of the problem by means of such methods of analysis of ill-posed problems as the method of elliptic regularization (vanishing viscosity method) and the quasisolutions method.
We investigate the behavior of the sequence of regularized semigroups $\{ e^{-i\mathbf L_nt},t>0\}$ depending on the choice of regularization $\{\mathbf L_n\}$ of the generating operator $\mathbf L$.
When there are no convergent sequences of regularized solutions, we study the convergence of the corresponding sequence of the regularized density operators.

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English version:
Journal of Mathematical Sciences, 2016, 213:3, 287–459

Bibliographic databases:

UDC: 517.946+517.98

Citation: V. Zh. Sakbaev, “Cauchy problem for degenerating linear differential equations and averaging of approximating regularizations”, Partial differential equations, CMFD, 43, PFUR, M., 2012, 3–172; Journal of Mathematical Sciences, 213:3 (2016), 287–459

Citation in format AMSBIB
\Bibitem{Sak12}
\by V.~Zh.~Sakbaev
\paper Cauchy problem for degenerating linear differential equations and averaging of approximating regularizations
\inbook Partial differential equations
\serial CMFD
\yr 2012
\vol 43
\pages 3--172
\publ PFUR
\publaddr M.
\mathnet{http://mi.mathnet.ru/cmfd207}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3086726}
\transl
\jour Journal of Mathematical Sciences
\yr 2016
\vol 213
\issue 3
\pages 287--459
\crossref{https://doi.org/10.1007/s10958-016-2719-z}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84955325408}


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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, “Razrushenie reshenii zadachi Koshi dlya nelineinogo uravnenii Shredingera”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(30) (2013), 159–171  mathnet  crossref
    2. V. Zh. Sakbaev, “Gradient blow-up of solutions to the Cauchy problem for the Schrödinger equation”, Proc. Steklov Inst. Math., 283 (2013), 165–180  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    3. 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
    4. 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
    5. I. V. Volovich, V. Zh. Sakbaev, “Universal boundary value problem for equations of mathematical physics”, Proc. Steklov Inst. Math., 285 (2014), 56–80  mathnet  crossref  crossref  isi  elib  elib
    6. Sakbaev V.Zh., Smolyanov O.G., Shamarov N.N., “Non-Gaussian Lagrangian Feynman-Kac Formulas”, Dokl. Math., 90:1 (2014), 416–418  crossref  mathscinet  zmath  isi  elib  scopus
    7. 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
    8. 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
    9. L. S. Efremova, V. Zh. Sakbaev, “Notion of blowup of the solution set of differential equations and averaging of random semigroups”, Theoret. and Math. Phys., 185:2 (2015), 1582–1598  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    10. V. Zh. Sakbaev, “On the law of large numbers for compositions of independent random semigroups”, Russian Math. (Iz. VUZ), 60:10 (2016), 72–76  mathnet  crossref  mathscinet  isi  elib  elib
    11. Yu. N. Orlov, V. Zh. Sakbaev, O. G. Smolyanov, “Unbounded random operators and Feynman formulae”, Izv. Math., 80:6 (2016), 1131–1158  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    12. V. Zh. Sakbaev, I. V. Volovich, “Self-adjoint approximations of the degenerate Schrödinger operator”, p-Adic Numbers Ultrametric Anal. Appl., 9:1 (2017), 39–52  crossref  mathscinet  zmath  isi  scopus
    13. I. V. Volovich, V. Zh. Sakbaev, “On quantum dynamics on $C^*$-algebras”, Proc. Steklov Inst. Math., 301 (2018), 25–38  mathnet  crossref  crossref  isi  elib  elib
    14. 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
    15. Sakbaev V.Zh., European Conference - Workshop Nonlinear Maps and Applications, Journal of Physics Conference Series, 990, IOP Publishing Ltd, 2018  crossref  mathscinet  isi  scopus
  • Современная математика. Фундаментальные направления
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