One-dimensional perturbations of selfadjoint operators with finite or discrete spectrum

  • Authors:

  • Yu. M. Arlinskiǐ;S. Hassi;H. S. V. de Snoo;E. R. Tsekanovskiǐ

  • Affiliations:

  • Department of Mathematical Analysis, East Ukrainian National University, Kvartal Molodyozhny, Lugansk, Ukraine;Department of Mathematics and Statistics, University of Vaasa, Vaasa, Finland;Department of Mathematics, University of Groningen, Groningen Nederland;Department of Mathematics, Niagara University, Niagara, NY

  • Venue:
  • Contemporary mathematics
  • Year:
  • 2001

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Abstract

There are many classical results concerning the spectrum of perturbations of selfadjoint operators; in particular the behaviour under "small perturbations" has been investigated throughoutly. In the present paper attention is paid to the asymptotic behaviour of eigenvalues under rank one perturbations when such perturbations become infinitely large. This leads in a natural way to the study of the eigenvalues of a selfadjoint limiting perturbation, which is necessarily a selfadjoint relation (multivalued linear operator). The use of extension theory and associated Q-functions facilitates the study of the asymptotic properties under such "large perturbations" and leads to similar interpretations as are known in the case of "small perturbations".