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
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
