7. Eigenvalues

Question 1

Simple Eigenvalue Problem in Matrix Form

Answer 1

A x = λ x

• this is satisfied for any λ when x=0
• it can only be satisfied when x≠0 for certain special values of λ, the eigenvalues λn with corresponding (non-zero) eigenvectors xn

Question 2

Simplest Eigenvalue Problem for a Second-Order Differential Operator
Equation & BCs

Answer 2

• d²y/dx² = λy

- with y(0) = y(π) = 0

Question 3

Simplest Eigenvalue Problem for a Second-Order Differential Operator
Solution

Answer 3

-for λ0:
y = Acos(x√λ) + Bsin(x√λ)
-sub in BCs, if y(0)=0 => A=0
-if y(π)=0 then for non-trivial solutions we require B≠0 so:
sin(x√λ)=0
=> √λ = n or λ=n²
-where n is a positive integer
=> y = sin(nx)

Question 4

Mathematical Setting

Sturm-Louisville Eigenvalue Problems

Answer 4

-d/dx( p(x) dy/dx) + q(x)y = λ w(x) y
-with y(a)=y(b)=0 and p(x), q(x) & w(x) given
-it is customary to consider systems with:
p(x) > 0
q(x) ≥ 0
w(x) > 0

Question 5

Sturm-Louisville Eigenvalue Problems

Eigenvalues

Answer 5

• all eigenvalues λ are real and positive

- there are infinitely many eigenvalues, we write 0

Question 6

Sturm-Louisville Eigenvalue Problems

Eigenfunctions

Answer 6

• eigenfunctions can be taken to be real
• eigenfunctions are orthogonal with respect to the weight function w(x)
• the eigenfunctions {yn(x)} form a complete set for functions satisfying the same boundary conditions

Question 7

Sturm-Louisville Eigenvalue Problems

Orthogonality of Eigenfunctions

Answer 7

-eigenfunctions are orthogonal with respect to the weight function w(x):
∫ w(x) ym(x) yn(x) dx = 0, m or Im m=n
-where Im = ∫ w ym² dx
-integrals between a and b

Question 8

Sturm-Louisville Eigenvalue Problems

Eigenfunctions as a Set for f(x)

Answer 8

f(x) = Σfj yj(x)
-sum from j=1 to j=∞
=>
∫ w(x) ym(x) f(x) dx = Σ ∫ w(x) ym(x) f(x) dx
=>
fm = 1/Im ∫ w(x) ym(x) f(x) dx

Question 9

Differential Operator

Simple Solution by Perturbation Analysis

Answer 9

-expand y = yo(x) + ε y1(x) + ε² y2(x) + …, ε«1 and sub into the equation and boundary conditions
-and specify the normalisation condition:
yo’(x) + ε y1’(x) + … = 1
-evaluate O(1) terms, a simple eigenvalue problem, there are infinitely many eigenvalue / eigenfunction solutions, take the n=1 ‘ground state’ ones
-evaluate O(ε) terms, O(ε²) terms etc.

Question 10

Differential Operator

General Solution Form

Answer 10

-we suppose that p(x)=po(x)+εp1(x) and q(x)=qo(x)+εq1(x) with w(x) unperturbed
=>
(Lo + εL1)(yo+εy1+ε²y2+…) = w(λo+ελ1+ε²λ2+…)(yo+εy1+…)
-with:
Lo = -d/dx (po d/dx) + qo
L1 = -d/dx (p1 d/dx) + q1
-and we use y(x)=0 at x=a,b
=>
yo(x) + εy1(x) + ε²y2(x) + … = 0 at x=a,b

Question 11

Differential Operator

General Solution at O(1)

Answer 11

Lo yo = λo w yo, yo(a)=yo(b)=0
-assume that this can be solved with eigenvalues Λj and eigenfunctions Yj(x), j=1,2,3,…
Lo Yj = Λj w Yj
-we choose yo=Ym(x) and λo=Λm

Question 12

Differential Operator

General Solution at O(ε)

Answer 12

• take O(ε) terms, sub in yo=Ym(x) and λo=Λm from the O(1) terms
• multiply by Ym and integrate from a to b
• this allows the equation to be rewritten in terms of inner products
• rearrange for λ1
• to determine y1(x) write, y1(x)=ΣajYj(x)
• this time multiply the equation by Yj(x) and integrate from a to b to get expression for aj

Question 13

Differential Operator

General Solution at O(ε²)

Answer 13

• take the O(ε²) terms, sub in yo=Ym(x) and λo=Λm
• multiply the equation by Ym(x) and integrate from a to b
• rearrange for λ2

Question 14

Eigenvalues of Matrices

Problem

Answer 14

-consider:
(Ao + εA1) x = λx
-the unpertubed problem is then Ao Xj = Λj Xj for j=1,2,3,…

Question 15

Eigenvalues of Matrices

Assumptions

Answer 15

-assume that Ao is real and symmetric so that the eigenvalues, Λj, are real
-assume that the eigenvectors are normalised and orthogonal:
Xi . Xj = δij
-assume that all Λj are distinct

Question 16

Eigenvalues of Matrices

Inner Product

Answer 16

⟨x,y⟩ = x . y = x^T y
-then
⟨x,Aoy⟩ = x^T Ao y = (x^T Ao y)^T = y^T Ao^T x = ⟨y,Aox⟩
-for all real x and y, Ao is self-adjoint

Question 17

Eigenvalues of Matrices

Steps

Answer 17

-perturbation analysis:
(Ao+εA1)(Xm + εx1 + ε²x2 +…) = (Λm+ελ1+ε²λ2+…)(Xm+ εx1 + ε²x2 +…)
-consider O(1), O(ε), etc. terms
-at each order take the inner product with Xm to find the eigenvalue
-then the inner product with Xj to find the eigenfunction

Question 18

Eigenvalues of Matrices

O(1) Steps

Answer 18

-automatically satisfied

Question 19

Eigenvalues of Matrices

O(ε) Steps

Answer 19

• take the O(ε) terms
• take the inner product of the equation with Xj
• when m=j this is solvable for λ1
• when m≠j the λ1 term vanishes by orthogonality so we can find ⟨x1,Xj⟩
• expand x1 in terms of Xj and find coefficients

Question 20

Eigenvalues of Matrices

O(ε²) Steps

Answer 20

• find the O(ε²) terms
• take the inner product of the equation with Xm
• sub in results from previous O() calculations and rearrange for λ2

Question 21

Eigenvalues of Matrices

O(ε) Eigenvalue

Answer 21

λ1 = ⟨x1,A1 Xm⟩ / ⟨Xm,Xm⟩

-where ⟨Xm,Xm⟩=1

Question 22

Eigenvalues of Matrices

O(ε²) Eigenvalue

Answer 22

λ2 = Σ ⟨Xj,A1 Xm⟩ ⟨Xm, A1 Xj⟩ / (Λm - Λj)

-sum from j=1 to j=N, not including j=m