On some results in SturmLiouville 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 SturmLiouville 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/studentpapers/record/2517903
 author
 Beck, Alexis
 supervisor

 Erik WahlĂ©n ^{LU}
 organization
 course
 MATX01 20112
 year
 2011
 type
 M2  Bachelor Degree
 subject
 publication/series
 Bachelor's Theses in Mathematical Sciences
 report number
 LUNFMA40122011
 ISSN
 16546229
 other publication id
 2011:K7
 language
 English
 id
 2517903
 date added to LUP
 20141215 14:09:14
 date last changed
 20181011 16:23:06
@misc{2517903, abstract = {Sturm's oscillation theorem says that the zeros of the nth eigenfunction of a SturmLiouville 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 = {16546229}, language = {eng}, note = {Student Paper}, series = {Bachelor's Theses in Mathematical Sciences}, title = {On some results in SturmLiouville theory and their generalizations to higher dimensions}, year = {2011}, }