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TMF, 1981, Volume 46, Number 3, Pages 291–299 (Mi tmf2332)  

This article is cited in 5 scientific papers (total in 5 papers)

Representation of free solutions for Schrödinger equations with strongly singular concentrated potentials

Yu. M. Shirokov


Abstract: A new representation is obtained for the Schrödinger equation constructed and solved in the author's earlier paper [1] for three-dimensional motion of a particle in the field of a strongly singular concentrated potential. In the new representation, the Schrödinger equation becomes free but the wave functions of the bound states have exponential growth at infinity. The transition to the new representation is linear but contains a procedure of analytic continuation, which makes it a transformation that does not possess a kernel and does not exist in the complete Hilbert space. It is shown that, using the new representation, one can readily obtain the complete solution of the original Schrödinger equation. The new “free-solution representation” is used to obtain the complete solution to the quantum problem of the motion of a particle in the field of a centrally symmetric concentrated potential that acts in states with $l\ne0$. Positivity of the metric has not been verified for the obtained solution. The possibility of applying the method to the quantum problem of several bodies with concentrated two-body interactions is noted.

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English version:
Theoretical and Mathematical Physics, 1981, 46:3, 191–196

Bibliographic databases:

Received: 16.04.1980

Citation: Yu. M. Shirokov, “Representation of free solutions for Schrödinger equations with strongly singular concentrated potentials”, TMF, 46:3 (1981), 291–299; Theoret. and Math. Phys., 46:3 (1981), 191–196

Citation in format AMSBIB
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\by Yu.~M.~Shirokov
\paper Representation of free solutions for Schr\"odinger equations with strongly singular concentrated potentials
\jour TMF
\yr 1981
\vol 46
\issue 3
\pages 291--299
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=622509}
\zmath{https://zbmath.org/?q=an:0458.35085|0486.35070}
\transl
\jour Theoret. and Math. Phys.
\yr 1981
\vol 46
\issue 3
\pages 191--196
\crossref{https://doi.org/10.1007/BF01032724}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1981MN84000001}


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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. I. S. Tsirova, Yu. M. Shirokov, “Quantum delta-like potential acting in the $P$ state”, Theoret. and Math. Phys., 46:3 (1981), 203–206  mathnet  crossref  mathscinet  zmath  isi
    2. I. S. Tsirova, “Singular potentials in a problem with noncentral interaction”, Theoret. and Math. Phys., 51:3 (1982), 561–563  mathnet  crossref  isi
    3. G. K. Tolokonnikov, “Differential rings used in Shirokov algebras”, Theoret. and Math. Phys., 53:1 (1982), 952–954  mathnet  crossref  mathscinet  zmath  isi
    4. Yu. G. Shondin, “Generalized pointlike interactions in $R_3$ and related models with rational $S$ matrix II. $l=1$”, Theoret. and Math. Phys., 65:1 (1985), 985–992  mathnet  crossref  mathscinet  isi
    5. B. S. Pavlov, A. A. Shushkov, “The theory of extentions and zero-radius potentials with internal structure”, Math. USSR-Sb., 65:1 (1990), 147–184  mathnet  crossref  mathscinet  zmath
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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