This document introduces to Sturm-Liouville eigenvalue problems, regular, singular and periodic, to find eigenvalues and eigenfunctions of partial differential equations. In introduces the concept of orthogonality of eigenfunctions. Then it focuses on how to generate the generalised Fourier series ...
OrthogonalityaEigenvalueProblems
MOTIVATION recall that anyperiodic squareintegrable function on E it it can bewritten
as a Fourier Series
fix EAn Unix
with yo Yaks sinks Yak cosKx
L
with An Life Unix ex
This only works because the Un are ORTHOGONAL under the innerproduct operation
funkVmaxdx it8mn
Can other sets offunctionsbe used to represent generalfunctions fix as a series in thisway
RECAP LinearAlgebra
For symmetric nxn matrixA we have Ayn than for EIGENVALUES In and EIGENVECTORS In
Can show
1 In are real
2 If An dm then In am o eigenvectorarrespondind todistinct eigenvaluesare orthogonal
3 Normalised eigenvectors
form an ORTHNORMAL basisfor IR sit anyvectorb in IR can bewritten
as I Ejakkk
This all generalises to Hermitianmatrices to G
So the eigenvalue problem Ay ta can bethought as a mean
ofgenerating an orthonormal
basis In We want to generalise theseideasto differentialoperatorsacting onspaces
of eigenvectors
offunctions
DEFINITION
Given a real vector space V the operation L Viv IR is called an INNERPRODUCT
iff
1 Cf g g f A fig EV SYMME
HAIRY
2
cafth g d Cf Chig g fig he V Linearity
3 Lf f D E
f O PositiveDEFINITE
e
g Let V L functionspace of squareintegrable functions on E it it
Then
cfg jg ax is an innerproduct
Can we extend the concept ofsymmetry
ofmatricesto operators acting on a functionspace
, DEFINITION
An operator L V V is called SELFADJOINT wrt the innerproduct c iff
Lf g CfLg tf.gov
Recall a matrix MEIRMIR is symmetric orselfadjoint
iff Matty My VI y ER
L
e
g
Consider the eigenvalue problem
L O some L e
LlyL dy with gig and boundaryconditions
yea y for
1 is L selfadjoint wrt fig fgelx
Lpg sax j's If e
fit t
from B Cs
4
2 Can we solve the problem
dy yea yea O EXERCISE
Ey
Solutions are
yea sinking kiss with eigenvalue tic E
EIGENFUNCTION
ie an infinite set ofeigenvalues dieand associated eigenfunctions ye and note ye are
orthogonal wrt inner product defined earlier
sin sin ax if jfk
ye y g if j K
dy yea yea o
off
Iso thtd o
y AcostaBsintx
yea A O
yCL BsinAL O AL Kit HE KI ya BsinK
d o o Axt B
y y AL O L O NO non trivialsolution
yea B O yL
Ice y ly let the use y my y AsinhfextBasha
13 0 A O no nontrivial solution
ylo y L AsinhFL O
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