Advanced

On some results in Sturm-Liouville theory and their generalizations to higher dimensions

Beck, Alexis (2011) In Bachelor's Thesis in Mathematical Sciences MATX01 20112
Mathematics (Faculty of Sciences)
Abstract
Sturm's oscillation theorem says that the zeros of the nth eigenfunction of a Sturm-Liouville problem with separated boundary conditions divides the domain into n connected pieces. We present a proof of this theorem which partly uses a variational characterization of the eigenfunctions. We also show how the variational characterization can be adapted to study some properties of the eigenvalues. The generalization of Sturm's oscillation theorem to higher dimensions is only partly true; the zero set divides the domain into at most n connected parts. This is Courant's nodal theorem. In the second part of the thesis we construct examples showing that the number of parts can be strictly less than n.
Please use this url to cite or link to this publication:
author
Beck, Alexis
supervisor
organization
course
MATX01 20112
year
type
M2 - Bachelor Degree
subject
publication/series
Bachelor's Thesis in Mathematical Sciences
report number
LUNFMA-4012-2011
ISSN
1654-6229
other publication id
2011:K7
language
English
id
2517903
date added to LUP
2014-12-15 14:09:14
date last changed
2014-12-15 14:25:11
@misc{2517903,
  abstract     = {Sturm's oscillation theorem says that the zeros of the nth eigenfunction of a Sturm-Liouville problem with separated boundary conditions divides the domain into n connected pieces. We present a proof of this theorem which partly uses a variational characterization of the eigenfunctions. We also show how the variational characterization can be adapted to study some properties of the eigenvalues. The generalization of Sturm's oscillation theorem to higher dimensions is only partly true; the zero set divides the domain into at most n connected parts. This is Courant's nodal theorem. In the second part of the thesis we construct examples showing that the number of parts can be strictly less than n.},
  author       = {Beck, Alexis},
  issn         = {1654-6229},
  language     = {eng},
  note         = {Student Paper},
  series       = {Bachelor's Thesis in Mathematical Sciences},
  title        = {On some results in Sturm-Liouville theory and their generalizations to higher dimensions},
  year         = {2011},
}