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

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

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
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
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
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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
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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
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
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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
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
19. Kurasov P., Stenberg F., “On the Inverse Scattering Problem on Branching Graphs”, J. Phys. A-Math. Gen., 35:1 (2002), 101–121
20. Kurasov P., “H-N-Perturbations of Self-Adjoint Operators and Krein's Resolvent Formula”, Integr. Equ. Oper. Theory, 45:4 (2003), 437–460
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
22. Kurasov P., “Triplet Extensions I: Semibounded Operators in the Scale of Hilbert Spaces”, J. Anal. Math., 107 (2009), 251–286
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