Perturbation Theoryward/teaching/m605/every2_perturb.pdf · then perturbation theory says that for...

22
Appendix D Perturbation Theory D.1 Simple Examples Let A = 1 0 0 0 0 1 0 0 0 0 2 0 0 0 0 3 , B = 0 1 10 10 1 0 10 10 10 10 4 10 10 10 10 6 . (D.1) The eigenvalues of A are 1, 2 and 3, where λ = 1 has multiplicity 2. To find the eigenvalue of the perturbed matrix A + ǫB corresponding to the unperturbed eigenvalue λ 0 = 2, we use non-degenerate perturbation theory, so λ = λ 0 + ǫλ 1 + O ( ǫ 2 ) (D.2) =2+ ǫ x T 0 Bx 0 x T 0 x 0 + O ( ǫ 2 ) (D.3) =2+ ǫ 0 0 1 0 0 1 10 10 1 0 10 10 10 10 4 10 10 10 10 6 0 0 1 0 0 0 1 0 0 0 1 0 + O ( ǫ 2 ) (D.4) =2+4ǫ + O ( ǫ 2 ) , (D.5) noting that A = A T , so x 0 , the eigenvector of A T with eigenvalue 2, is the same as x 0 , the eigenvector of A with eigenvalue 2. We can do likewise for λ 0 = 3 to find λ =3+6ǫ + O ( ǫ 2 ) . (D.6) 203

Transcript of Perturbation Theoryward/teaching/m605/every2_perturb.pdf · then perturbation theory says that for...

Page 1: Perturbation Theoryward/teaching/m605/every2_perturb.pdf · then perturbation theory says that for ǫ small, the eigenvalues of A+ǫB are λ = 1±1ǫ 2+4ǫ 3+6ǫ +O ǫ2 (D.13) These

Appendix D

Perturbation Theory

D.1 Simple Examples

Let

A =

1 0 0 00 1 0 00 0 2 00 0 0 3

, B =

0 1 10 10−1 0 10 1010 10 4 1010 10 10 6

. (D.1)

The eigenvalues of A are 1, 2 and 3, where λ = 1 has multiplicity 2. To find theeigenvalue of the perturbed matrix A + ǫB corresponding to the unperturbedeigenvalue λ0 = 2, we use non-degenerate perturbation theory, so

λ = λ0 + ǫλ1 + O(

ǫ2)

(D.2)

= 2 + ǫx∗T

0 Bx0

x∗T0 x0

+ O(

ǫ2)

(D.3)

= 2 + ǫ

[

0 0 1 0]

0 1 10 10−1 0 10 1010 10 4 1010 10 10 6

0010

[

0 0 1 0]

0010

+ O(

ǫ2)

(D.4)

= 2 + 4ǫ + O(

ǫ2)

, (D.5)

noting that A = AT , so x∗0, the eigenvector of AT with eigenvalue 2, is the same

as x0, the eigenvector of A with eigenvalue 2. We can do likewise for λ0 = 3 tofind

λ = 3 + 6ǫ + O(

ǫ2)

. (D.6)

203

Page 2: Perturbation Theoryward/teaching/m605/every2_perturb.pdf · then perturbation theory says that for ǫ small, the eigenvalues of A+ǫB are λ = 1±1ǫ 2+4ǫ 3+6ǫ +O ǫ2 (D.13) These

For λ0 = 1, we must use degenerate perturbation theory, which says that

Bc = λ1Mc (D.7)

where

Bij ≡ v∗Ti Bvj Mij ≡ v∗T

i vj , (D.8)

so equation D.7 becomes[

0 1−1 0

]

c = λ1

[

1 00 1

]

c, (D.9)

hence λ1 is i or −i, so the perturbed eigenvalues for λ0 = 1 are

λ = 1 + ǫ

i−i

+ O(

ǫ2)

(D.10)

In summary, the eigenvalues of A + ǫB are

λ =

1 ± iǫ2 + 4ǫ3 + 6ǫ

+ O(

ǫ2)

(D.11)

These are compared to numerically calculated eigenvalues in figure D.1.If instead,

B =

1 0 10 100 −1 10 1010 10 4 1010 10 10 6

, (D.12)

then perturbation theory says that for ǫ small, the eigenvalues of A + ǫB are

λ =

1 ± 1ǫ2 + 4ǫ3 + 6ǫ

+ O(

ǫ2)

(D.13)

These are compared to numerically calculated eigenvalues in figure D.2.

D.2 Wilkinson’s Matrix

We now choose to look at an example where perturbation theory is very poorat approximating the perturbed eigenvalues. Following Wilkinson [31], we let

A =

20 2019 20 0

18 20. . .

. . .

0 2 201

. (D.14)

204

Page 3: Perturbation Theoryward/teaching/m605/every2_perturb.pdf · then perturbation theory says that for ǫ small, the eigenvalues of A+ǫB are λ = 1±1ǫ 2+4ǫ 3+6ǫ +O ǫ2 (D.13) These

0 0.005 0.010.98

1

1.02Real Component of First Eigenvalue

Rea

l Com

pone

nt

Size of Perturbation (ε)0 0.005 0.01

−0.01

0

0.01Imag. Component of 1st Eigenvalue

Imag

. Com

pone

nt

Size of Perturbation (ε)

0 0.005 0.010.98

1

1.02Real Component of Second Eigenvalue

Rea

l Com

pone

nt

Size of Perturbation (ε)0 0.005 0.01

−0.01

0

0.01Imag. Component of 2nd Eigenvalue

Imag

. Com

pone

nt

Size of Perturbation (ε)

0 0.005 0.011.98

2

2.02

2.04

Real Component of Third Eigenvalue

Rea

l Com

pone

nt

Size of Perturbation (ε)0 0.005 0.01

−0.01

0

0.01Imag. Component of 3rd Eigenvalue

Imag

. Com

pone

nt

Size of Perturbation (ε)

0 0.005 0.01

3

3.05

3.1Real Component of Fourth Eigenvalue

Rea

l Com

pone

nt

Size of Perturbation (ε)0 0.005 0.01

−0.01

0

0.01Imag. Component of 4th Eigenvalue

Imag

. Com

pone

nt

Size of Perturbation (ε)

calculated by MATLAB

predicted by perturbation theory

Figure D.1: The real (first column) and imaginary (second column) componentsof the four eigenvalues of A + ǫB, plotted versus ǫ, with A and B defined byequation D.1.

205

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0 0.005 0.010.98

1

1.02First Eigenvalue

Eig

enva

lue

Size of Perturbation (ε)

0 0.005 0.010.98

1

1.02Second Eigenvalue

Eig

enva

lue

Size of Perturbation (ε)

0 0.005 0.011.98

2

2.02

2.04

Third Eigenvalue

Eig

enva

lue

Size of Perturbation (ε)

0 0.005 0.01

3

3.05

3.1Fourth Eigenvalue

Eig

enva

lue

Size of Perturbation (ε)

calculated by MATLAB

predicted by perturbation theory

Figure D.2: The four eigenvalues (all real) of A+ ǫB, plotted versus ǫ, where Bis defined by equation D.12.

206

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This has eigenvalues 1, 2, . . . , 20 and the eigenvector corresponding to eigenvalueλ is

(−20)20−λ

(20−λ)!

...(−20)2

2!−2010...0

λ − 1 zeroes

(D.15)

The eigenvector of AT corresponding to eigenvalue λ is

0...0120

(20)2

2!...

(20)λ−1

(λ−1)!

20 − λ zeroes

(D.16)

D.2.1 Eigenvalues

Let

B =

0 0... . .

.

0 01 0 · · · 0

. (D.17)

Then perturbation theory says that the eigenvalues of A+ǫB are given, to orderǫ, by

λn = λ0n + ǫλ1n (D.18)

= n + ǫx∗T

0 Bx0

x∗T0 x0

(D.19)

= n + ǫ2019 (−1)

n

(20 − n)! (n − 1)!(D.20)

or, if we generalise A and B to be m by m matrices,

λn = n + ǫmm−1 (−1)

m−n

(m − n)! (n − 1)!(D.21)

207

Page 6: Perturbation Theoryward/teaching/m605/every2_perturb.pdf · then perturbation theory says that for ǫ small, the eigenvalues of A+ǫB are λ = 1±1ǫ 2+4ǫ 3+6ǫ +O ǫ2 (D.13) These

10−15

10−10

10−5

10

10.1

10.2

10.3

10.4

10.5

10.6

Size of Perturbation (ε)

Rea

l Com

pone

nt o

f Eig

enva

lue

Real Component of Tenth Eigenvalue

predicted range of validity of perturbation theorycalculated by MATLABpredicted by perturbation theory

Figure D.3: The 10th eigenvalue in the m = 20 case, as a function of ǫ, ascomputed numerically by MATLAB. The eigenvalue predicted by perturbationtheory is also shown. The ‘predicted range’ gives the number η for which wemust have ǫ ≪ η in order for perturbation theory to be valid, as in equationD.22.

where this is valid for

ǫmm−1

(m − n)! (n − 1)!≪ 1. (D.22)

For example, when m = 50, perturbation theory predicts

λ8 = 8 + ǫ · 2.5085 × 1028 (D.23)

so we expect this to be valid for ǫ ≪ 10−28.The eigenvalues in the m = 20 case, as a function of ǫ, were computed

numerically in MATLAB and are shown in figures D.4–D.9. Also, the 10theigenvalues in the m = 20 case, as a function of ǫ, is compared to that predictedby perturbation theory. This is shown in figure D.3.

Error in Computation

Since the Wilkinson matrix has such poorly behaved eigenvalues, we expectthat numerical computations of its perturbed eigenvalues will be difficult. Toillustrate this, we compute the eigenvalues of the perturbed Wilkinson matrixfor different values of m and ǫ using two different programs, MATLAB andMathematica. The calculations were performed to 15 digits. The result is shown

208

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10−15

10−10

10−5

−5

0

5

10

15

20

25Real Component of Eigenvalues

Rea

l Com

pone

nt o

f Eig

enva

lues

Size of Perturbation (ε)0 10 20

−10

−5

0

5

10Eigenvalues

Imag

inar

y C

ompo

nent

of E

igen

valu

es

Real Component of Eigenvalues

10−15

10−10

10−5

−10

−5

0

5

10Imaginary Component of Eigenvalues

Imag

inar

y C

ompo

nent

of E

igen

valu

es

Size of Perturbation (ε)

predicted range of validity of perturbation theorycalculated by MATLAB

Figure D.4: The eigenvalues in the m = 20 case, as a function of ǫ, as computednumerically by MATLAB. The ‘predicted range’ gives the number η, as a func-tion of λ, for which we must have ǫ ≪ η in order for perturbation theory to bevalid, as in equation D.22.

209

Page 8: Perturbation Theoryward/teaching/m605/every2_perturb.pdf · then perturbation theory says that for ǫ small, the eigenvalues of A+ǫB are λ = 1±1ǫ 2+4ǫ 3+6ǫ +O ǫ2 (D.13) These

−100

1020

30

−10−5

05

1010

−15

10−10

10−5

Real Component

Eigenvalues

Imaginary Component

Siz

e of

Per

turb

atio

n (ε)

Figure D.5: A 3D representation of the eigenvalues in the m = 20 case, as afunction of ǫ, as computed by MATLAB.

210

Page 9: Perturbation Theoryward/teaching/m605/every2_perturb.pdf · then perturbation theory says that for ǫ small, the eigenvalues of A+ǫB are λ = 1±1ǫ 2+4ǫ 3+6ǫ +O ǫ2 (D.13) These

Size of Perturbation (ε)

Siz

e of

Mat

rix

10−50

10−40

10−30

10−20

10−10

10

15

20

25

30

35

40

45

50

1e−14

1e−12

1e−10

1e−08

1e−06

1e−04

1e−02

1

1e+02

Difference Between Mathematica and MATLAB EigenvaluesRegion Where Eigenvalues Become All Real

Figure D.6: The difference between the computed eigenvalues of A+ ǫB accord-ing to MATLAB and Mathematica, as a function of m and ǫ. The dashed linesindicate the region in which the eigenvalues, according to Mathematica, becomeentirely real.

211

Page 10: Perturbation Theoryward/teaching/m605/every2_perturb.pdf · then perturbation theory says that for ǫ small, the eigenvalues of A+ǫB are λ = 1±1ǫ 2+4ǫ 3+6ǫ +O ǫ2 (D.13) These

Size of Perturbation (ε)

Siz

e of

Mat

rix

10−50

10−40

10−30

10−20

10−10

10

15

20

25

30

35

40

45

50

1e−14

1e−12

1e−10

1e−08

1e−06

1e−04

1e−02

1

1e+02

Difference Between Mathematica and Perturbation Theory EigenvaluesRegion Where Eigenvalues Become All Real

Figure D.7: The difference between the computed eigenvalues of A+ ǫB accord-ing to Mathematica and the predicted eigenvalues from perturbation theory, asa function of m and ǫ. The dashed lines indicate the region in which the eigen-values, according to Mathematica, become entirely real. This also correspondsto the approximate boundary of the region of validity of perturbation theory.

212

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Size of Perturbation (ε)

Siz

e of

Mat

rix

10−50

10−40

10−30

10−20

10−10

10

15

20

25

30

35

40

45

50

1e−14

1e−12

1e−10

1e−08

1e−06

1e−04

1e−02

1

1e+02

Difference Between MATLAB and Perturbation Theory EigenvaluesRegion Where Eigenvalues Become All Real

Figure D.8: The difference between the computed eigenvalues of A+ ǫB accord-ing to MATLAB and the predicted eigenvalues from perturbation theory, as afunction of m and ǫ. The dashed lines indicate the region in which the eigen-values, according to Mathematica, become entirely real. This also correspondsto the approximate boundary of the region of validity of perturbation theory.

213

Page 12: Perturbation Theoryward/teaching/m605/every2_perturb.pdf · then perturbation theory says that for ǫ small, the eigenvalues of A+ǫB are λ = 1±1ǫ 2+4ǫ 3+6ǫ +O ǫ2 (D.13) These

in figure D.6. More precisely, the plot is of the maximum of the absolute valueof the complex difference between the eigenvalues as computed by MATLABand Mathematica. To determine which is most accurate, we also plot each ascompared with perturbation theory (figures D.7 and D.8). Observe that, withinthe region of validity of perturbation theory, Mathematica agrees very well withperturbation theory. We assume then that the error depicted in figure D.7 isdue to errors from perturbation theory, not Mathematica.

D.2.2 Sherman-Woodbury-Morrison Formula

If we wish to solve the related problem Ax + ǫBx = b, with A and B as above,then perturbation theory predicts

x =(

A−1 − A−1ǫBA−1)

b. (D.24)

Since for our choice of B, we can write B = uvT , then we can apply the Sherman-Woodbury-Morrison formula which says that the exact solution is

x =

(

A−1 − A−1ǫBA−1

1 + ǫvT A−1u

)

b, (D.25)

so we expect perturbation theory to be valid for ǫvT A−1u ≪ 1.Now

A−1 =

19!20! (−20) 18!

20! (−20)2 17!20! · · · (−20)19 0!

20!

18!19! (−20) 17!

19!

17!18!

0. . .

0!1!

(D.26)

so

vT A−1u =(−20)19

20!(D.27)

for m = 20. We can generalise this for any m to obtain

vT A−1u =(−m)m−1

m!. (D.28)

We then expect perturbation theory to be valid for

ǫ ≪(

vT A−1u)−1

(D.29)

ǫ ≪(

(−m)m−1

m!

)−1

. (D.30)

For example, when m = 50, this means that perturbation theory is valid forǫ ≪ 2 × 10−19.

214

Page 13: Perturbation Theoryward/teaching/m605/every2_perturb.pdf · then perturbation theory says that for ǫ small, the eigenvalues of A+ǫB are λ = 1±1ǫ 2+4ǫ 3+6ǫ +O ǫ2 (D.13) These

−100

1020

30

−10−5

05

1010

−15

10−10

10−5

100

Real Component

Imaginary Component

Siz

e of

Per

turb

atio

n

pseudospectrumeigenvalues of perturbed matrix

Figure D.9: A 3D version of the pseudospectrum for the unperturbed matrix(the surface), in the m = 20 case, in comparison with the eigenvalues of theperturbed matrix (the lines), as computed by MATLAB. The lines in this figureare identical to those in figure D.5.

D.2.3 Pseudospectrum

Following Embree and Trefethen [7], we define the pseudospectrum of A to be

Λǫ(A) = z ∈ C : z ∈ Λ (A + E) for some E with ‖E‖ ≤ ǫ (D.31)

where Λ (A + E) is the set of eigenvalues of the matrix (A + E) and ‖ · ‖ is amatrix norm induced by a vector norm. We choose ‖ · ‖ to be the 2-norm. Since‖ǫB‖ = ǫ, then the eigenvalues of the perturbed matrix A + ǫB are elementsof Λǫ(A). The pseudospectrum of A in the m = 20 case is shown in figuresD.9 and D.10. Note that the pseudospectrum of A agrees very well with theeigenvalues of A + ǫB as computed by MATLAB, although the eigenvalues ofA + ǫB are all strictly within the boundary predicted by the pseudospectrum.This is especially visible in figure D.10. This suggests that there exist somematrices C with ‖C‖ = 1 such that the eigenvalues of A + ǫC are slightlyfurther away from the unperturbed eigenvalues than the eigenvalues of A + ǫB.

215

Page 14: Perturbation Theoryward/teaching/m605/every2_perturb.pdf · then perturbation theory says that for ǫ small, the eigenvalues of A+ǫB are λ = 1±1ǫ 2+4ǫ 3+6ǫ +O ǫ2 (D.13) These

Real Component

Imag

inar

y C

ompo

nent

0 5 10 15 20−5

−4

−3

−2

−1

0

1

2

3

4

5

eigenvalues of unperturbed matrixeigenvalues of perturbed matrixcontour plot of pseudospectrum

−13

−12

−11

−10

−9

−8

−7

−6

−5

−4

Figure D.10: The pseudospectrum for the unperturbed matrix, in the m = 20case.

216

Page 15: Perturbation Theoryward/teaching/m605/every2_perturb.pdf · then perturbation theory says that for ǫ small, the eigenvalues of A+ǫB are λ = 1±1ǫ 2+4ǫ 3+6ǫ +O ǫ2 (D.13) These

D.3 Domain Perturbations

Consider now the first order corrections to the eigenvalues of a differential equa-tion where the domain has been perturbed by some perturbation of order ǫ. Welook in specific at the problem

∆φ + λφ = 0 in Ωǫ (D.32a)

φ = 0 on δΩǫ (D.32b)

whereδΩǫ : R = 1 + ǫf (θ) (D.33)

and

f (θ) =

∞∑

n=−∞

aneinθ, a−n = an. (D.34)

We will now find the first order corrections to the eigenvalues by two differentmethods, and show that they produce the same result.

D.3.1 General Method

For the first method, we start by writing

φ = φ0 + ǫφ1 + ǫ2φ2 + . . . (D.35)

λ = λ0 + ǫλ1 + ǫ2λ2 + . . . (D.36)

and equating terms of equal order from equation D.32a, we obtain that

O (1) : ∆φ0 + λ0φ0 = 0 (D.37)

O (ǫ) : ∆φ1 + λ0φ1 = −λ1φ0 (D.38)

L (φ) = −λ1φ0. (D.39)

Performing a Taylor expansion of the boundary conditions,

0 = φ (1 + ǫf (θ) , θ) (D.40)

= φ (1, θ) + ǫf (θ) φ,r (1, θ) +ǫ2f2 (θ)

2φ,rr (1, θ) (D.41)

and equating terms of equal order,

O (1) : 0 = φ0 (1, θ) (D.42)

O (ǫ) : 0 = φ1 (1, θ) + f (θ) φ0,r (1, θ) (D.43)

217

Page 16: Perturbation Theoryward/teaching/m605/every2_perturb.pdf · then perturbation theory says that for ǫ small, the eigenvalues of A+ǫB are λ = 1±1ǫ 2+4ǫ 3+6ǫ +O ǫ2 (D.13) These

First Eigenvalue

The unperturbed eigenfunction of the first eigenvalue is

φ0 = J0 (j01r) (D.44)

where jmk is the kth positive zero of the Bessel function Jm. Now we know that

(φ0,L (φ1)) − (φ1,L (φ0)) =

∂D

(φ0φ1,r − φ1φ0,r) ds (D.45)

where

(f, g) ≡∫

D

fg dx. (D.46)

Using equations D.37, D.39, D.42 and D.43, equation D.45 becomes

−λ1 (φ0, φ0) =

∂D

f (θ) (φ0,r)2

ds (D.47)

so

λ1 = −∫

∂Df (θ) (φ0,r)

2ds

D(φ0)

2dx

(D.48)

= −∫ θ=2π

θ=0

∑∞n=−∞

(

aneinθ)

(J ′0 (j01) j01)

2dθ

∫ θ=2π

θ=0dθ∫ r=1

r=0r (J0 (j01r))

2dθ

(D.49)

= − (J ′0 (j01))

2j201

∫ θ=2π

θ=0

(

a0 +∑∞

n=1

(

aneinθ + a−ne−inθ))

2π 12 (J ′

0 (j01))2 (D.50)

= −j201

2

∫ θ=2π

θ=0

(

a0 +∞∑

n=1

(2Re (an) cos (nθ) +

2 Im (an) sin (nθ))

)

dθ (D.51)

= −2j201a0. (D.52)

The first eigenvalue is then

λ = j201 − ǫ2j2

01a0 + O(

ǫ2)

. (D.53)

Second and Third Eigenvalues

The second and third eigenvalues of the unperturbed system are the same, sowe have

φ0 = c1v1 + c2v2 (D.54)

where c1 and c2 are arbitrary constants and

v1 = J1 (j11r) cos (θ) , v2 = J1 (j11r) sin (θ) . (D.55)

218

Page 17: Perturbation Theoryward/teaching/m605/every2_perturb.pdf · then perturbation theory says that for ǫ small, the eigenvalues of A+ǫB are λ = 1±1ǫ 2+4ǫ 3+6ǫ +O ǫ2 (D.13) These

As before, we can use

(v1,L (φ1)) − (φ1,L (v1)) =

∂D

(v1φ1,r − φ1v1,r) ds, (D.56)

which, upon applying equations D.37, D.39, D.42 and D.43, becomes

−λ1 (v1, φ0) =

∂D

f (θ) φ0,rv1,r ds (D.57)

λ1c1 (v2, v1) − λ1c2 (v2, v2) = c1

∂D

f · v21,r ds + c2

∂D

f · v1,rv2,r ds. (D.58)

We can do likewise, with v1 in equation D.56 replaced with v2 to obtain

λ1c2 (v2, v1) − λ1c2 (v2, v2) = c1

∂D

f · v1,rv2,r ds + c2

∂D

f · v22,r ds. (D.59)

We can express these two equations together as

[ ∫

f · v21,r

f · v1,rv2,r∫

f · v1,rv2,r

f · v22,r

] [

c1

c2

]

= −λ1

[

(v1, v1) (v1, v2)(v1, v2) (v2, v2)

] [

c1

c2

]

.

(D.60)Using the definition of v1 and v2 from equation D.55 and the result that

∫ r=1

r=0

r (J1 (j11r))2dr = −J0 (j11) J2 (j11)

2=

1

2(J0 (j11))

2, (D.61)

this simplifies to

[ ∫

f · v21,r

f · v1,rv2,r∫

f · v1,rv2,r

f · v22,r

] [

c1

c2

]

= −π

2(J0 (j11))

2λ1

[

c1

c2

]

. (D.62)

If we re-write this as[

A BB C

] [

c1

c2

]

= Q

[

c1

c2

]

(D.63)

then the eigenvalues are given by

Q =A + C ±

√A2 + C2 − 2AC + 4B2

2, (D.64)

219

Page 18: Perturbation Theoryward/teaching/m605/every2_perturb.pdf · then perturbation theory says that for ǫ small, the eigenvalues of A+ǫB are λ = 1±1ǫ 2+4ǫ 3+6ǫ +O ǫ2 (D.13) These

where

A =

∂D

f (θ) (v1,r (1, θ))2

ds (D.65)

=

∫ θ=2π

θ=0

(

∞∑

n=−∞

aneinθ

)

(J ′1 (j11) j11 cos (θ))

2dθ (D.66)

= (J ′1 (j11))

2j211

∫ θ=2π

θ=0

(

a0 +

∞∑

n=1

(

aneinθ + a−ne−inθ)

)

cos2 (θ) dθ (D.67)

= (J ′′0 (j11))

2j211

∫ θ=2π

θ=0

(

a0 +

∞∑

n=1

(2Re (an) cos (nθ)

+2 Im (an) sin (nθ))

)

cos2 (θ) dθ (D.68)

= (J ′′0 (j11))

2j211π (Re (a2) + a0) (D.69)

≡ A0 + ξ. (D.70)

We likewise obtain that

C =

∂D

f (θ) (v2,r (1, θ))2

ds (D.71)

=

∫ θ=2π

θ=0

(

∞∑

n=−∞

aneinθ

)

(J ′1 (j11) j11 sin (θ))

2dθ (D.72)

= (J ′′0 (j11))

2j211π (−Re (a2) + a0) (D.73)

= −A0 + ξ (D.74)

and

B =

∂D

f (θ) v1,r (1, θ) v2,r (1, θ) ds (D.75)

=

∫ θ=2π

θ=0

(

∞∑

n=−∞

aneinθ

)

(J ′1 (j11) j11)

2sin (θ) cos (θ) dθ (D.76)

= − (J ′′0 (j11))

2j211π Im (a2) , (D.77)

so that

Q =A + C ±

√A2 + C2 − 2AC + 4B2

2(D.78)

= ξ ±√

A20 + B2. (D.79)

220

Page 19: Perturbation Theoryward/teaching/m605/every2_perturb.pdf · then perturbation theory says that for ǫ small, the eigenvalues of A+ǫB are λ = 1±1ǫ 2+4ǫ 3+6ǫ +O ǫ2 (D.13) These

From the definition of Q, we obtain that

−λ1π

2(J0 (j11))

2= Q (D.80)

−λ1π

2(J0 (j11))

2= (J ′′

0 (j11))2j211a0π ± (J ′′

0 (j11))2j211π |a2| (D.81)

λ1 = 2j211 (± |a2| − a0) (D.82)

So the second eigenvalue is

λ = j211 + ǫ2j2

11 (− |a2| − a0) + O(

ǫ2)

(D.83)

and the third isλ = j2

11 + ǫ2j211 (|a2| − a0) + O

(

ǫ2)

. (D.84)

D.3.2 Method of Assumed Solution

For this method, we follow Wolf and Keller [32]. We start by assuming that thesolution is of the form

φ (r, θ, ǫ) =∞∑

n=−∞

An (ǫ) Jn (kr) einθ (D.85)

where

An (ǫ) = δ|n|mαn + ǫβn + ǫ2γn + . . . (D.86)

R (θ, ǫ) = 1 + ǫ∑

n

aneinθ + ǫ2∑

n

bneinθ + O(

ǫ3)

(D.87)

√λ ≡ k = k0 + ǫk1 + ǫ2k2 + . . . (D.88)

Applying boundary conditions,

φ (R (θ, ǫ) , θ, ǫ) =∑

n

An (ǫ) Jn (kR (θ, ǫ)) einθ = 0. (D.89)

Expanding this in a Taylor series, and writing kR = k0 + (k − k0) + k (R − 1),we obtain

n

An

Jn (k0) + J ′n (k0) (k − k0 + k (R − 1)) +

1

2J ′′

n (k0) (k − k0 + k (R − 1))2

+ . . .

einθ = 0. (D.90)

Applying equations D.86–D.88 and equating terms of equal order, we obtainthat

O (1) :(

αmeimθ + α−me−imθ)

Jm (k0) = 0 (D.91)

O (ǫ) :(

α−mJ ′−me−imθ + αmJ ′

meimθ)

(

k1 + k0

l

aleilθ

)

+

n

βnJneinθ = 0. (D.92)

221

Page 20: Perturbation Theoryward/teaching/m605/every2_perturb.pdf · then perturbation theory says that for ǫ small, the eigenvalues of A+ǫB are λ = 1±1ǫ 2+4ǫ 3+6ǫ +O ǫ2 (D.13) These

From equation D.91, we obtain that k0 = jmp, so m = 0, p = 1 corresponds tothe first eigenvalue and m = 1, p = 1 will give the second and third eigenvalues.

First Eigenvalue

Under the condition that m = 0, equation D.92 becomes

0 = 2α0J′0

(

k1 + k0

l

aleilθ

)

+∑

n

βnJneinθ. (D.93)

Equating terms that have θ = 0, we obtain

0 = 2α0J′0 (j01) (k1 + k0a0) + β0J0 (j01) (D.94)

0 = 2α0J′0 (j01) (k1 + k0a0) (D.95)

0 = k1 + k0a0, (D.96)

sok1

k0= −a0 (D.97)

and

λ = k2 (D.98)

= k20 + ǫ2k0k1 + O

(

ǫ2)

(D.99)

= j201 − ǫ2j2

01a0 + O(

ǫ2)

, (D.100)

which agrees with equation D.53.

Second and Third Eigenvalues

Again following Wolf and Keller [32], we equate the coefficients of eimθ in equa-tion D.92 to obtain

0 = α−mJ ′−m (k0) k0a2m + αmJ ′

m (k0) (k1 + k0a0) + βmJm (k0) (D.101)

0 = α−mJ ′−m (k0) k0a2m + αmJ ′

m (k0) (k1 + k0a0) (D.102)

0 = α−mk0a2m + αm (k1 + k0a0) (D.103)

andk1

k0= −αma2m

αm− a0. (D.104)

Alternately, equating the coefficients of e−imθ in equation D.92, we obtain

k1

k0= −αma2m

αm− a0. (D.105)

Equating equations D.104 and D.105, we obtain that∣

αma2m

αm

eiω =

αma2m

αm

e−iω (D.106)

222

Page 21: Perturbation Theoryward/teaching/m605/every2_perturb.pdf · then perturbation theory says that for ǫ small, the eigenvalues of A+ǫB are λ = 1±1ǫ 2+4ǫ 3+6ǫ +O ǫ2 (D.13) These

where

ω = arg

(

αma2m

αm

)

(D.107)

= − arg

(

αma2m

αm

)

(D.108)

so that

eiω = e−iω (D.109)

eiω = ±1. (D.110)

It then follows that

k1

k0= ±

αma2m

αm

− a0 (D.111)

= ± |a2m| − a0, (D.112)

so the second and third eigenvalues are

λ = j211 + ǫ2j2

11 (± |a2| − a0) + O(

ǫ2)

(D.113)

which agrees with equations D.83 and D.84.

D.3.3 Result

Similarly, Wolf and Keller [32] compute the second order correction to the firsteigenvalue. Then under the condition that the area of the domain does notchange under the perturbation (which causes a0 = 0, among other things), theyobtain that

AλII = πj211−

π1/2j211

j201

[

AλI − πj201

1 + j01J ′2 (j01) /J2 (j01)

]1/2

+O[

AλI − πj201

]

. (D.114)

where A is the area of the domain, and λI and λII are the first and secondeigenvalues, respectively. Normalising A to unity, we obtain the plot shown infigure D.11.

223

Page 22: Perturbation Theoryward/teaching/m605/every2_perturb.pdf · then perturbation theory says that for ǫ small, the eigenvalues of A+ǫB are λ = 1±1ǫ 2+4ǫ 3+6ǫ +O ǫ2 (D.13) These

0 20 40 60 80 1000

20

40

60

80

First Eigenvalue (λI)

Sec

ond

Eig

enva

lue

(λ II

)

Figure D.11: The first and second eigenvalues for perturbations to a circle.When there are no perturbations, λI is at a minimum.

224