Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-27T16:34:19.069Z Has data issue: false hasContentIssue false
  • English
  • Français

A generalisation of Niessen's limit-circle criterion

Published online by Cambridge University Press:  14 February 2012

Christer Bennewitz
Christer Bennewitz
Affiliation:
Department of Mathematics, University of Uppsala
Get access Rights & Permissions [Opens in a new window]

Synopsis

Let S and T be formally symmetric ordinary differential operators defined on a real interval I. It is assumed that the order of S is constant and everywhere strictly higher than the possibly varying order of T. The main result of this paper (Theorem 2.3) gives necessary and sufficient conditions for maximality of the deficiency indices of the differential relation Su = Tv considered in a Hilbert space with a scalar product which is a Dirichlet integral (see section 2) belonging to S. The conditions generalise those given in [5] for less general choices of operators S and T. For certain choices of Dirichlet integral they are explicit integrability conditions on the coefficients of the Dirichlet integral and the operator T.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 78 , Issue 1-2 , 1977 , pp. 81 - 90
Copyright
Copyright © Royal Society of Edinburgh 1977

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Bennewitz, C.Spectral theory for pairs of differential operators. Ark. Mat. 15 (1977).CrossRefGoogle Scholar
2Hille, E.Lectures on ordinary differential operators (Reading, Mass.: Addison-Wesley, 1969).Google Scholar
3Karlsson, B.Generalization of a theorem of Everitt. J. London Math. Soc. 9 (1974), 131–141.CrossRefGoogle Scholar
4Karlsson, B.On the limit circle case for pairs of ordinary differential operators in the left positive case. Inst. Mittag-Leffler Rep. 7 (1975).Google Scholar
5Niessen, H. D. A necessary and sufficient limit-circle criterion for left-definite eigenvalue problems. Conf. Theor. Ordinary and Partial Differential Equations, Univ. Dundee, 1974. Lecture Notes in Mathematics 280 (Berlin: Springer, 1974).Google Scholar
6Pleijel, Å.Spectral theory for pairs of formally selfadjoint ordinary differential operators. J. Indian Math. Soc. 34 (1970), 259–268.Google Scholar
7Pleijel, Å. A positive symmetric ordinary differential operator combined with one of lower order. Spectral theory and asymptotics of differential equations. Proc. Conf. Scheveningen, 1973. Math. Studies 13 (Amsterdam: North Holland, 1974).Google Scholar
8Schneider, A. and Niessen, H. D.Linksdefinite singuläre kanonische Eigenwertprobleme I. J. Reine Angew. Math. 281 (1976), 13–52.Google Scholar