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A - Historical remarks

Published online by Cambridge University Press:  05 November 2011

S. Albeverio
Affiliation:
Rheinische Friedrich-Wilhelms-Universität Bonn
P. Kurasov
Affiliation:
Stockholms Universitet
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Summary

Extension theory for symmetric operators

Extension theory and Nevanlinna R-functions

Probably the most interesting example zof a singular rank one perturbation is the perturbation of the boundary condition for a second order ordinary differential operator. This is one of the first mathematical problems where the extension theory plays an indispensable role. In 1909–1910 H. Weyl investigated in his famous papers [954, 955] the behavior of the solutions to the second order differential equation under variation of the boundary condition. He was also the first to ask the question: How does the spectrum change under such a perturbation? He proved that the absolutely continuous spectrum is invariant under such a perturbation. The question by H.Weyl concerning other types of spectrum has been investigated by F. Wolf [964], N. Aronszajn [97], and N.Aronszajn and W.F.Donoghue [99]. See also the paper by V.A. Javrian [493]. So it was H.Weyl who was the first to understand the importance of this class of perturbations from the mathematical point of view. The first mathematically rigorous investigation of singular perturbations of partial differential operator was carried out by F.A.Berezin and L.D.Faddeev [135]. These authors have shown that such perturbations can be described using the extension theory of symmetric operators. This paper was extremely important because it clarified the relation between partial differential operators with point interactions and Krein's formula describing the resolvents of all self–adjoint extensions of a given symmetric operator.

Type
Chapter
Information
Singular Perturbations of Differential Operators
Solvable Schrödinger-type Operators
, pp. 327 - 352
Publisher: Cambridge University Press
Print publication year: 2000

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  • Historical remarks
  • S. Albeverio, Rheinische Friedrich-Wilhelms-Universität Bonn, P. Kurasov, Stockholms Universitet
  • Book: Singular Perturbations of Differential Operators
  • Online publication: 05 November 2011
  • Chapter DOI: https://doi.org/10.1017/CBO9780511758904.010
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  • Historical remarks
  • S. Albeverio, Rheinische Friedrich-Wilhelms-Universität Bonn, P. Kurasov, Stockholms Universitet
  • Book: Singular Perturbations of Differential Operators
  • Online publication: 05 November 2011
  • Chapter DOI: https://doi.org/10.1017/CBO9780511758904.010
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Historical remarks
  • S. Albeverio, Rheinische Friedrich-Wilhelms-Universität Bonn, P. Kurasov, Stockholms Universitet
  • Book: Singular Perturbations of Differential Operators
  • Online publication: 05 November 2011
  • Chapter DOI: https://doi.org/10.1017/CBO9780511758904.010
Available formats
×