On some results in Sturm-Liouville theory and their generalizations to higher dimensions
(2011) In Bachelor's Theses in Mathematical Sciences MATX01 20112Mathematics (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:
http://lup.lub.lu.se/student-papers/record/2517903
- author
- Beck, Alexis
- supervisor
-
- Erik Wahlén LU
- organization
- course
- MATX01 20112
- year
- 2011
- type
- M2 - Bachelor Degree
- subject
- publication/series
- Bachelor's Theses 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
- 2018-10-11 16:23:06
@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 Theses in Mathematical Sciences}}, title = {{On some results in Sturm-Liouville theory and their generalizations to higher dimensions}}, year = {{2011}}, }