Arkiv för Matematik

  • Ark. Mat.
  • Volume 7, Number 6 (1969), 543-550.

Some remarks about the limit point and limit circle theory

Åke Pleijel

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Abstract

LetL be a formally selfadjoint differential operator and p a real-valued function, both on ax<∞. The deficiency indices are the numbers of solutions of Lupu for Im λ>0 and for Im λ<0 which have a certain regularity at x=∞. (A) If p(x)≥0 this regularity means that the integral of p(x)u2 converges at infinity. (B) If p changes its sign for arbitrarily large values of x but L has a positive definite Dirichlet integral it is natural to relate the regularity to this integral. Weyl’s classical study of the deficiency indices is reviewed for (A) with the help of elementary theory of quadratic forms. Individual bounds are found for the deficiency indices also when L is of odd order. It is then indicated how the method carriers over to (B).

Article information

Source
Ark. Mat., Volume 7, Number 6 (1969), 543-550.

Dates
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485893692

Digital Object Identifier
doi:10.1007/BF02590893

Mathematical Reviews number (MathSciNet)
MR240378

Zentralblatt MATH identifier
0281.34014

Rights
1969 © Almqvist & Wiksell

Citation

Pleijel, Åke. Some remarks about the limit point and limit circle theory. Ark. Mat. 7 (1969), no. 6, 543--550. doi:10.1007/BF02590893. https://projecteuclid.org/euclid.afm/1485893692


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References

  • Everitt, W. N., Singular differential equations. I. The even order case. Math. Ann. 156 (1964), 9–24.
  • Glazman, I. M., On the theory of singular differential operators. Uspehi Mat. Nauk (N.S.)5, no. 6 (40) (1950), 102–135. Also in A. M. S. Translations, series 1, vol. 4: Differential Equations, pp. 331–372.
  • Weyl, H., Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen. Math. Ann. 68 (1910), 220–269. Also in Selecta Hermann Weyl. Birkhäuser Verlag, Basel and Stuttgart, 1956.