Skip to main content

Green's functions for pairs of formally selfadjoint ordinary differential operators

Invited Lectures

Part of the Lecture Notes in Mathematics book series (LNM,volume 280)

Abstract

A spectral theory is deduced for differential eigenvalue problems related to a formally selfadjoint differential equation Su=λ Tu, where u is complex-valued and λ is the eigenvalue parameter. The equation or rather the differential operators S and T are considered on an arbitrary open interval of the real axis. The lower order operator T is assumed to have a positive definite Dirichlet integral which serves as scalar product in spectral theorems determined by symmetric boundary conditions. The theory is given in terms of ordered pairs u/\(\dot u\)of functions. Thus symmetric boundary conditions are certain subrelations of {u/\(\dot u\): Su=T\(\dot u\)}. If for instance T is the identity operator the boundary conditions are equally well described as conditions on u only. As far as the spectral theorem is concerned the method of the paper is easily transferred to the case when S instead of T has a positive definite Dirichlet integral. In the here considered case with T positive a kernel representation of the resolvent is deduced and used to prove the regularity of the elements of eigenspaces belonging to finite intervals of the spectral axis. The theory was worked out independently of the investigations by F.W. Schäfke, A. Schneider and H.-D. Niesser of systems of first order equations to which it seems related in different respects.

Keywords

  • Continuous Derivative
  • Spectral Theorem
  • Finite Dimensional Subspace
  • Linear Hull
  • Symmetric Boundary Condition

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   46.00
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. Bennewitz, Chr. and Pleijel, Åke. Selfadjoint extension of ordinary differential operators. To appear in the Proceedings of the Colloquium on Mathematical Analysis at Jyväskylä, Finland, August 17–21, 1970. (Mimeographed preprint available from the Department of Mathematics at Uppsala University.)

    Google Scholar 

  2. Niessen, Heinz-Dieter. Singuläre S-hermitesche Rand-Eigenwertprobleme. Manuscripta Mathematica, 3, 35–68, 1970. (Contains references to earlier papers by F.W. Schäfke and A. Schneider.)

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. Pleijel, Åke. Generalization of the spectral theory of ordinary formally self-adjoint differential operators. To appear in Journal of the Indian Mathematical Society, 34, 1972. (Mimeographed preprint available from the Department of Mathematics at Uppsala University.)

    Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1972 Springer-Verlag

About this paper

Cite this paper

Pleijel, Å. (1972). Green's functions for pairs of formally selfadjoint ordinary differential operators. In: Everitt, W.N., Sleeman, B.D. (eds) Conference on the Theory of Ordinary and Partial Differential Equations. Lecture Notes in Mathematics, vol 280. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0066924

Download citation

  • DOI: https://doi.org/10.1007/BFb0066924

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-05962-2

  • Online ISBN: 978-3-540-37618-7

  • eBook Packages: Springer Book Archive