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Mat. Sb. (N.S.), 1988, Volume 136(178), Number 2(6), Pages 163–177 (Mi msb1734)  

This article is cited in 22 scientific papers (total in 22 papers)

Boundary conditions on thin manifolds and the semiboundedness of the three-particle Schrödinger operator with pointwise potential

B. S. Pavlov


Abstract: The purpose of this article is to describe the formulation of a selfadjoint spectral problem with boundary conditions on a sufficiently thin manifold. Namely, let $\mathscr L$ be a selfadjoint operator in $L_2(\mathbf R^n)$, let $L$ be a smooth manifold, let $\mathscr L_0$ be the restriction of $\mathscr L$ to the lineal in $\mathscr D(\mathscr L_0)$ consisting of all functions which vanish in a neighborhood of $L$.
It is shown that the deficiency elements of this restriction can be represented as “tensor layers” with densities of a definite class of smoothness, concentrated on the “boundary” of $L$. If $L$ is sufficiently thin, there is only one family of deficiency elements, and it is analogous to the single-layer potentials. In this case, calculation of the boundary form and the description of the selfadjoint extensions appears to be quite simple. This case is studied in detail because the investigation of the simplest model of the three-particle problem of quantum mechanics reduces to it.
Bibliography: 16 titles.

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English version:
Mathematics of the USSR-Sbornik, 1989, 64:1, 161–175

Bibliographic databases:

UDC: 517.9
MSC: Primary 35J10; Secondary 35P20, 81F10
Received: 14.05.1987

Citation: B. S. Pavlov, “Boundary conditions on thin manifolds and the semiboundedness of the three-particle Schrödinger operator with pointwise potential”, Mat. Sb. (N.S.), 136(178):2(6) (1988), 163–177; Math. USSR-Sb., 64:1 (1989), 161–175

Citation in format AMSBIB
\Bibitem{Pav88}
\by B.~S.~Pavlov
\paper Boundary conditions on thin manifolds and the semiboundedness of the three-particle Schr\"odinger operator with pointwise potential
\jour Mat. Sb. (N.S.)
\yr 1988
\vol 136(178)
\issue 2(6)
\pages 163--177
\mathnet{http://mi.mathnet.ru/msb1734}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=954922}
\zmath{https://zbmath.org/?q=an:0687.35064}
\transl
\jour Math. USSR-Sb.
\yr 1989
\vol 64
\issue 1
\pages 161--175
\crossref{https://doi.org/10.1070/SM1989v064n01ABEH003300}


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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. Slobodin V., “Homogeneous Boundary-Value-Problems for Polyharmonic Operator with Boundary-Condition on Thin Sets”, 306, no. 5, 1989, 1051–1055  mathscinet  zmath  isi
    2. B. S. Pavlov, A. V. Strepetov, “Exactly solvable model of electron scattering by an inhomogeneity in a thin conductor”, Theoret. and Math. Phys., 90:2 (1992), 152–156  mathnet  crossref  mathscinet  isi
    3. A. A. Kiselev, B. S. Pavlov, N. N. Penkina, M. G. Suturin, “Allowance for interaction symmetry in the theory of extensions”, Theoret. and Math. Phys., 91:2 (1992), 453–461  mathnet  crossref  mathscinet  isi
    4. Cheremshantsev S., “Hamiltonians with Zero-Range Interactions Supported by a Brownian Path”, Ann. Inst. Henri Poincare-Phys. Theor., 56:1 (1992), 1–25  mathscinet  zmath  isi
    5. A. K. Motovilov, “Algebraic version of extension theory for a quantum system with internal structure”, Theoret. and Math. Phys., 97:2 (1993), 1217–1228  mathnet  crossref  mathscinet  zmath  isi
    6. K. A. Makarov, V. V. Melezhik, A. K. Motovilov, “The point interactions in the problem of three quantum particles with internal structure”, Theoret. and Math. Phys., 102:2 (1995), 188–207  mathnet  crossref  mathscinet  zmath  isi
    7. Antonets M. Geyler V., “A Quasi-Two-Dimensional Charged Particle in a Tilted Magnetic Field: Asymptotic Properties of the Spectrum”, Russ. J. Math. Phys., 3:4 (1995), 413–422  mathscinet  zmath  isi
    8. Yu. G. Shondin, “Semibounded local hamiltonians for perturbations of the laplacian supported by curves with angle points in $\mathbb R^4$”, Theoret. and Math. Phys., 106:2 (1996), 151–166  mathnet  crossref  crossref  mathscinet  zmath  isi
    9. K. A. Makarov, V. V. Melezhik, “The Efimov effect and collaps in three-body systems with point-like interactions. I”, Theoret. and Math. Phys., 107:3 (1996), 755–769  mathnet  crossref  crossref  mathscinet  zmath  isi
    10. P. Kurasov, “Energy Dependent Boundary Conditions and the Few-Body Scattering Problem”, Rev. Math. Phys, 09:07 (1997), 853  crossref
    11. Yu. A. Kuperin, S. B. Levin, “Extension theory approach to scattering and annihilation in the $\bar pd$ system”, Theoret. and Math. Phys., 118:1 (1999), 60–76  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    12. Korovina M., “On the Schrodinger Equation with Potential Concentrated on a Surface”, Differ. Equ., 36:9 (2000), 1279–1283  mathnet  crossref  mathscinet  zmath  isi
    13. Kurasov P., Pavlov B., “Few-Body Krein's Formula”, Operator Theory and Related Topics, Operator Theory : Advances and Applications, 118, eds. Adamyan V., Gohberg I., Gorbachuk M., Gorbachuk V., Kaashoek M., Langer H., Popov G., Birkhauser Verlag Ag, 2000, 225–254  mathscinet  zmath  isi
    14. Korovina M., “The Construction of a Self-Adjoint Extension of the Schrodinger Operator with Potential Concentrated on a Stratified Pencil of Planes”, Differ. Equ., 37:6 (2001), 829–833  mathnet  crossref  mathscinet  zmath  isi
    15. Exner P., Ichinose T., “Geometrically Induced Spectrum in Curved Leaky Wires”, J. Phys. A-Math. Gen., 34:7 (2001), 1439–1450  crossref  mathscinet  zmath  adsnasa  isi
    16. Albeverio S., Kurasov P., “Singular Cluster Interactions in Few-Body Problems”, Evolution Equations and their Applications in Physical and Life Sciences, Lecture Notes in Pure and Applied Mathematics, 215, eds. Lumer G., Weis L., Marcel Dekker, 2001, 277–292  mathscinet  zmath  isi
    17. Kurasov P., Watanabe K., “On H-4-Perturbations of Self-Adjoint Operators”, Partial Differential Equations and Spectral Theory, Operator Theory : Advances and Applications, 126, eds. Demuth M., Schulze B., Birkhauser Verlag Ag, 2001, 179–196  mathscinet  zmath  isi
    18. Korovina M., “The Construction of a Self-Adjoint Extension of the Schro-Dinger Operator with Potential Concentrated on a Pencil of Planes: I”, Differ. Equ., 38:6 (2002), 816–829  mathnet  crossref  mathscinet  zmath  isi
    19. Kurasov P., Stenberg F., “On the Inverse Scattering Problem on Branching Graphs”, J. Phys. A-Math. Gen., 35:1 (2002), 101–121  crossref  mathscinet  zmath  adsnasa  isi
    20. Kurasov P., “H-N-Perturbations of Self-Adjoint Operators and Krein's Resolvent Formula”, Integr. Equ. Oper. Theory, 45:4 (2003), 437–460  crossref  mathscinet  zmath  isi
    21. Kurasov P., Posilicano A., “Finite Speed of Propagation and Local Boundary Conditions for Wave Equations with Point Interactions”, Proc. Amer. Math. Soc., 133:10 (2005), 3071–3078  crossref  mathscinet  zmath  isi  elib
    22. Kurasov P., “Triplet Extensions I: Semibounded Operators in the Scale of Hilbert Spaces”, J. Anal. Math., 107 (2009), 251–286  crossref  mathscinet  zmath  isi
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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