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Perturbation of differential operators on high-codimension manifold and the extension theory for symmetric linear relations in an indefinite metric space

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Abstract

The problem of realization of nontrivial perturbations supported on thin sets of “codimension” ν in Rn for elliptic operators of order m, when ν≥2m, is formulated as one of construction of the self-adjoint extensions of some symmetric linear relation in an indefinite metric space. The self-adjoint extensions and their resolvents are described. It is found that the same extensions can be obtained as a result of extensions of some symmetric operator in L2 (Rn) with it going out to a larger indefinite metric space. But such an operator is chosen already by the “nonlocal” boundary conditions. Applications to quantum models of point interactions are discussed.

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References

  1. Pavlov B.S.,The extension theory and explicitly soluble models. Uspekhi Mat. Nauk.42 (1988), no. 6, 99–131.

    Google Scholar 

  2. Shirokov Y.M.,Algebra of three-dimensional generalized functions. Teor. Mat. Fiz.40 (1979), no. 3, 348–354.

    Google Scholar 

  3. Albeverio S., Gestezy F., Hoegh-Krohn R., Holden H.:Solvable models in quantum mechanics, Springer, New York, 1988.

    Google Scholar 

  4. Hepp K.,Theorie de la renormalization Springer, New York, 1969.

    Google Scholar 

  5. Berezin F.A.,To the Lee model., Mat. Sborn.60, (1963), no. 4, 425–453.

    Google Scholar 

  6. Shondin Yu.G., Quantum mechanical models in Rn connected to the extensions of the energy operator in Pontrjagin space., Teor. Mat. Fiz.74 (1988), no. 3, 331–344.

    Google Scholar 

  7. Krein M.G., Langer H.,About defect subspaces and generalized resolvents of Hermitian operator in the space IIκ., Funkt. Anal. i Priloz.5 (1971), no. 2, 59–71; no. 3, 54–69.

    Google Scholar 

  8. Krein M.G., Langer H.,Uber die Q-function eines π-hermitischen operator in raume IIκ., Acta Sci. Math.34 (1973), 191–230.

    Google Scholar 

  9. Arens R.,Operational calculus of linear relations., Pacific J. Math.11 (1961), no. 1, 9–23.

    Google Scholar 

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Authors
  1. Yu. G. Shondin
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Additional information

In Memory of Mikhail Constantinovich Polivanov

V. A. Steklov Mathematical Institute, Russian Academy of Sciences. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 3, pp. 466–472, September, 1992.

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Shondin, Y.G. Perturbation of differential operators on high-codimension manifold and the extension theory for symmetric linear relations in an indefinite metric space. Theor Math Phys 92, 1032–1037 (1992). https://doi.org/10.1007/BF01017080

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  • DOI: https://doi.org/10.1007/BF01017080

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