[PPT] - An inverse nodal problem for two-parameter Sturm-Liouville systems PowerPoint Presentation

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An inverse nodal problem for two-parameter Sturm-Liouville systems

Bruce A. Watson joint work with Paul Binding

School of Mathematics University of the Witwatersrand Johannesburg South Africa

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Introduction

For the two-parameter system y′′

j + (λaj + µbj + qj)yj = 0,

n

[0, 1], (1)

f Sturm-Liouville equations, linked by the eigen-parameters

(λ, µ), with (for simplicity of presentation) boundary conditions yj(0) = 0, (2) yj(1) = 0, j = 1, 2, (3) where aj, bj ∈ C1, a′

j, b′ j ∈ AC and qj ∈ L1 for j = 1, 2, we show

that a single sequence of pairs of nodal points, one of the pair for each eigenfunction yjk, j = 1, 2, k = 1, 2, ..., suffices to determine qj, j = 1, 2, uniquely.

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Klein’s oscillation theorem

The oscillation theory for the eigenfunctions of the system (1, 2, 3) has be well studied, and yields a natural indexing of the eigenvalues. Theorem Assume < δ(t1, t2) :=

a1(t1)

b1(t1) a2(t2) b2(t2)

,

for all (t1, t2) ∈ [0, 1]2, (4) and that n = (n1, n2) is a pair of non-negative integers. Then there exists a unique eigenvalue pair (λn, µn) of (1) - (3) so that the corresponding eigenfunctions yj have, respectively, nj zeros in (0, 1) for j = 1, 2.

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Definiteness - 1

From the definiteness assumption (4), there exist constants c, c > 0 so that c ≤ δ(t1, t2) ≤ c, for all t1, t2 ∈ [0, 1]. It follows from the work of Faierman that we may assume < a1(t1), b2(t2), a2(t2), b1(t1), for all t1, t2 ∈ [0, 1], (5) after a nonsingular change of λ, µ axes (although for simplicity we retain the original λ, µ notation).

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Definiteness - 2

Lemma There exist m1, m2 ∈ N, such that 1 √a1 1 √a2 > m1 m2 > 1 √b1 1 √b2 , where m1, m2 are even.

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Definiteness - 3

Proof. From (4),

a1(t1)b2(t2) > a2(t2)b1(t1), for all t1, t2 ∈ [0, 1]. Since a1, a2, b1, b2 are all positive, we may take square roots of both sides of the above inequality to yield

a1(t1)
b2(t2) >
a2(t2)
b1(t1).

Integrating this inequality with respect to t1 and then with respect to t2 gives 1

a1(t1) dt1

1

b2(t2) dt2 >

1

a2(t2) dt2

1

b1(t1) dt1.

Hence 1 √a1 1 √a2 > 1 √b1 1 √b2 from which the lemma follows directly.

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Eigencurves

Several authors, including Richardson, have studied two-parameter problems via eigencurve methods. Standard results on parametric dependence then allow us to consider the nth eigenvalue µjn of the jth problem in (1)-(3) as a continuous function of λ. The graph of this function is called the nth eigencurve for problem j. The eigenvalue pair (λn, µn) of Theorem 1 is at the intersection of the njth eigencurves for problems j = 1 and j = 2.

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Eigencurve asymptotics

Richardson gave eigencurve asymptotics. Turyn considered the “asymptotic directions", (γ, 1), of an eigencurve, i.e. if λ/µn(λ) → γ as λ → ∞ along the eigencurve. Binding and Browne, Faierman and Rynne also considered eigenvalue asymptotics but most give them to an order less than we need, or only for more restricted coefficients,

r are for (λn, µn) with one of the nj fixed, corresponding to

eigenvalue pairs along a fixed eigencurve. Faierman gave asymptotics of the type we need, but with implicit coefficients varying over an interval, whereas we require explicit coefficients. Our approach depends instead on using (4) to select a

ne-parameter subset of eigenvalue pairs (λk, µk) with a

special asymptotic direction (γ, 1).

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Asymptotic direction - 1

The following corollary to the lemma gives a key step towards finding an asymptotic direction along which eigenfunction and eigenvalue approximations can easily be found, as opposed to along eigencurves. We write fj(τ)[t] :=

τaj(t) + bj(t),

j = 1, 2, (6) Corollary If m1, m2 are as in Lemma 2, then there exists γ > 0 for which m2 1 f1(γ) = m1 1 f2(γ).

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Asymptotic direction - 2

Proof. Observe from Lemma 2 that

m2 1 f1(0) = m2 1

b1 < m1

1

b2 = m1

1 f2(0). Also from Lemma 2 it follows that m2 lim

γ→∞

1 f1(γ) √γ = m2 1 √a1 > m1 1 √a2 = m1 lim

γ→∞

1 f2(γ) √γ . Let g(γ) = 1 [m2f1(γ) − m1f2(γ)]. Then g(0) < 0 while for sufficiently large γ > 0, g(γ) > 0. Consequently, since the fj are continuous, there exists γ > 0 for which g(γ) = 0.

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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One-parameter family

For k = 0, 1, 2, . . . let ω(k) = (ω1(k), ω2(k)) where ωj(k) := mj

k + 1

2

− 1,

j = 1, 2. We consider the subset of (λn, µn) defined by λk := λω(k), (7) µk := µω(k). (8) It will be seen that the leading terms in the asymptotics of λk, µk are given by ¯ λk, ¯ µk where ¯ µk =

m1(k + 1

2)π

1

0 f1(γ)

2 =

m2(k + 1

2)π

1

0 f2(γ)

2 , (9) ¯ λk = γ¯ µk. (10) Thus (λk, µk) have asymptotic direction (γ, 1).

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Notation

For (λ, µ) = (λk, µk) denote yj by yk

j , we write

σjk = (λkaj + µkbj)1/4, ¯ σjk = (¯ λkaj + ¯ µkbj)1/4 = (¯ µk)1/4 fj(γ), ρj(t, λ, µ) = t

λaj + µbj,

ρk

j (t)

= ρj(t, λk, µk) = t σ2

jk,

¯ ρk

j (t)

= ρj(t, ¯ λk, ¯ µk) = t ¯ σ2

jk.

Observed that there exists a constant c > 0, depending only on a1, b1, a2, b2, such that min

t∈[0,1] ¯

σjk ≥ c √ k, for all k ∈ N, j = 1, 2.

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Solution asymptotics - 1

Theorem For (λ, µ) ∈ Sn as n → ∞, where Sn is a family of non-empty subsets of S with inf{σ2

j (t) | t ∈ [0, 1], (λ, µ) ∈ Sn} ≥ n,

for all n ∈ N, if yj(t) is the solution of (1) with yj(0) = 0 and y′

j(0) = 1,

yj(t) = 1 σj(0)σj

sin ρj +

t q∗

j − qj

σ2

j

sin[ρj(t) − ρj] sin ρj dτ + O 1 n2

y′(t)

= σj(t) σj(0)

cos ρj −

σ′

j

σ3

j

sin ρj + t q∗

j − qj

σ2

j

cos[ρj(t) − ρj(τ)] sinj ρ dτ + O 1 n2

.

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Solution asymptotics - 2

Here q∗

j

= σ′′

j

σj − 2 σ′

j

σj 2 , (11) ¯ qj = q∗

j − qj,

(12) ¯ Qj = 1 ¯ qj fj(γ), (13) for (λ, µ) ∈ S := {(λ, µ) | λ, µ ∈ R, λ, µ > 0}, (14) then sup

(λ,µ)∈S

q∗L1 < ∞. (15) Note that q∗(t) depends on both λ and µ.

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Nodal points

Let 0 < xk

j (1) < xk j (2) < · · · < xk j (ωj(k)) < 1 denote the nodal

points (zeros in (0, 1)) of the eigenfunction yk

j corresponding to

the eigenvalue (λk, µk). Define tk

j (n), ξk j (n) and ζk j (n) via the equations

ρk

j (tk j (n))

= nπ, ρk

j (ξk j (n))

=

n − 3

4

π,

ρk

j (ζk j (n))

=

n − 1

4

π.

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Eigenvalue and nodal asymptotics - 1

Theorem In the notation of (9), (10) we have λk = ¯ λk + O(1), µk = ¯ µk + O(1), as k → ∞. The nodal points of yk

j are given by

xk

j (n) = tk j (n) + O

1 k2

,

n = 1, . . . , ωj(k).

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Eigenvalue and nodal asymptotics - 2

Proof. If M > 0 and ˆ

λk ∈ [¯ λk − Mk, ¯ λk + Mk] and ˆ µk ∈ [¯ µk − Mk, ¯ µk + Mk] for each sufficiently large k ∈ N, then yj(1, ˆ λk, ˆ µk) = 1 ¯ σjk(0)¯ σjk(1)

sin ¯

ρj(1, ˆ λk, ˆ µk) + O 1 k

,

yj′(1, ˆ λk, ˆ µk) = ¯ σjk(1) ¯ σjk(0)

cos ¯

ρj(1, ˆ λk, ˆ µk) + O 1 k

.

Thus the first order approximation to the eigencondition (3) becomes sin ρj(1, ˆ λk, ˆ µk) = O 1 k

,

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Eigenvalue and nodal asymptotics - 3

which gives ρj(1, ˆ λk, ˆ µk) = π(1 + ωj(k)) + O 1 k

.

Substituting back the definition of ωj we obtain ρj(1, ˆ λk, ˆ µk) = πmj

k + 1

2

+ O

1 k

,

j = 1, 2. With δk = ˆ λk − ¯ λk and ǫk = ˆ µk − ¯ µk, the two-variable Taylor expansions of ρ1(1, ˆ λk, ˆ µk) and ρ2(1, ˆ λk, ˆ µk) give   1

a1 ¯ σ2

1k

1

b1 ¯ σ2

1k

1

a2 ¯ σ2

2k

1

b2 ¯ σ2

2k

  δk ǫk

=

O 1 k

(16)

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Eigenvalue and nodal asymptotics - 4

which from the definition of ¯ λk and ¯ µk leads to

1

πm1(k+ 1

2)

1 πm2(k+ 1

2)

M

δk ǫk

=

O 1 k

.

(17) Here M := A1 B1 A2 B2

(18)

where we write Aj = 1 aj fj(γ), Bj = 1 bj fj(γ), D = A1B2 − A2B1. (19)

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Eigenvalue and nodal asymptotics - 5

Here D = det M > 0 by the definiteness condition, and so there are unique δk, ǫk = O(1) such that (ˆ λk, ˆ µk) is an eigenvalue, for large k. Some careful analysis now shows that yj(t) := yj(t, ˆ λk, ˆ µk) has precisely ωj(k) zeros in (0, 1) and that they are placed

ne in each of the intervals [ζk

j (i), ξk j (i + 1)], i = 1, . . . , ωj(k),

i.e., xk

j (i) ∈ [ζk j (i), ξk j (i + 1)], i = 1, . . . , ωj(k).

Now yj(xk

j (i)) = 0 giving sin ρk j (xk j (i)) = O

1

k

, or more

precisely ρk

j (xk j (i)) = πi + O

1

k

.

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Eigenvalue and nodal asymptotics - 6

Hence πi + O 1 k

= ρk

j (xk j (i)) =

xk

j (i)

σ2

jk =

xk

j (i)

¯ σ2

jk + O

1 k

giving

xk

j (i)

fj(γ) = πi √¯ µk + O 1 k2

.

Thus xk

j (i)

tk

j (i)

fj(γ) = O 1 k2

.

Since there exist constants k2 > k1 > 0 so that k1 ≤ fj(γ)(t) ≤ k2, for all t ∈ [0, 1], we find that tk

j (i) − xk j (i) = O

1 k2

.

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Eigenvalue asymptotics

Theorem As k → ∞ we have λk = ¯ λk + A + o(1), µk = ¯ µk + B + o(1), where A and B are given by A = B2 ¯ Q1 − B1 ¯ Q2 D , B = − ¯ Q1A2 − A1 ¯ Q2 D . We remark that if the condition q ∈ L1 is strengthened to q ∈ AC then the o(1) term in the above theorem can be improved to O(1/k).

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Density - 1

Lemma If

v(k)

k

k∈N is a sequence of positive integers with

v(k) < k, k ∈ N, and

v(k)

k

k∈N is dense in [0, 1], then each of the

sequences of nodal points (xk

j (mjv(k)))k, j = 1, 2, is dense in

[0, 1]. Note that an algorithm for choosing a sequence (v(k))k∈N with 0 < v(k) < k such that

v(k)

k

k∈N is dense in [0, 1] can be found

in McLaughlin’s work on the inverse nodal problem.

Proof. By Theorem 5,

xk

j (n) = tk j (n) + O

1 k2

,

n = 1, . . . , ωj(k). Hence it suffices to prove the density of the sequences (tk

1(v(k)))k and (t2(v(k)))k in [0, 1].

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Density - 2

With ϕj(t) =

R t

0 fj(γ)

R 1

0 fj(γ) there are constants c0 and c1 so that

0 < c0 ≤ γaj(t) + bj(t) ≤ c1 for all t ∈ [0, 1] and j = 1, 2, making ϕj a homeomorphism of [0, 1]. Now nπ = ρk

j (tk j (n)) = mj

k + 1

2

πϕj(tk

j (n)) + O

1 k

,

j = 1, 2, and in particular ϕj(tk

j (mjv(k))) = v(k)

k + 1

2

+ O 1 k

= v(k)

k + O 1 k

.

Since

v(k)

k

k∈N is dense in [0, 1], (ϕj(tk

j (mjv(k))))k is dense in

[0, 1]. Finally, as ϕj is a homeomorphism on [0, 1] it follows that (tk

j (mjv(k)))k is dense in [0, 1].

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Uniqueness - 1

Theorem Let (v(k))k∈N be a sequence of positive integers with v(k) < k and

v(k)

k

k∈N is dense in [0, 1]. Then the sequence

(xk

1(m1v(k)), xk 2(m2v(k)))k, of nodal points and knowledge of

which of αj and βj − π are zero, j = 1, 2, uniquely determine the boundary conditions αj, βj ∈ [0, π), and hj := qj − aj B2Q1 − B1Q2 D + bj A2Q1 − A1Q2 D , j = 1, 2, where q1, q2 ∈ L1(0, 1), and we have used the notation of (6), (19) and Qj = 1 qj fj(γ).

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Uniqueness - 2

Proof. If the potentials qj in (1) are replaced by ˜ qj ∈ L1(0, 1), then denote the corresponding nodal points by ˜ xk

j (n), the

eigenvalues by (˜ λk, ˜ µk) and the solutions as in Theorem 4 by ˜ yk

j , j = 1, 2.

Here ˜ γ = γ, ˜ mj = mj, j = 1, 2, we have ωj = ˜ ωj in (7) which ensures that the relabelling of the eigenvalues as given in (7) and (8) is the same for both problems. Suppose ˜ xk

j (mjv(k)) = xk j (mjv(k)), k ∈ N, j = 1, 2.

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Uniqueness - 3

We begin by observing that yk

j ˜

yk

j (t)

= 1 (¯ σk

j (0)¯

σk

j (t))2

sin ρk

j (t) sin ˜

ρk

j (t) + O

1 k

. (20)

Let x ∈ [0, 1] and (xki

j (mjv(ki)))i be a subsequence of

(xk

j (mjv(k)))k which converges to x. For this subsequence we

have = lim

i→∞ ¯

µ1

ki

xki

j (mjv(ki))

[(˜ λki − λki)aj + (˜ µki − µki)bj + ˜ qj − qj]yki

j ˜

yki

j

= lim

i→∞

x [(˜ A − A)aj + (˜ B − B)bj + ˜ qj − qj]¯ µ1

kiyki j ˜

yki

j ,

where the sequence (¯ µ1

kiyki j ˜

yki

j )i is uniformly bounded.

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Uniqueness - 4

Hence 0 = lim

i→∞

x (˜ A − A)aj + (˜ B − B)bj + ˜ qj − qj fj(γ) zki (21) where zki = sin ρk

j (t) sin ˜

ρk

j (t).

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Uniqueness - 5

Now σ2

jk = ¯

σ2

jk + O

¯

µ−1/2

k

giving

sin ρk

j sin ˜

ρk

j

= 1 2 − 1 2 cos

2√¯

µk t fj(γ)

+ O
¯

µ−1/2

k

,

cos ρk

j cos ˜

ρk

j

= 1 2 + 1 2 cos

2√¯

µk t fj(γ)

+ O
¯

µ−1/2

k

.

Applying the Riemann-Lebesgue Lemma to (21) with the above simplifications of zk we have 0 = x (˜ A − A)aj + (˜ B − B)bj + ˜ qj − qj fj(γ) . (22)

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Uniqueness - 6

Differentiating (22) we obtain 0 = (˜ A − A)aj + (˜ B − B)bj + ˜ qj − qj, for j = 1, 2, (23) almost everywhere in [0, 1]. Recall that ˜ A − A = −B2(˜ Q1 − Q1) − B1(˜ Q2 − Q2) D , ˜ B − B = A2(˜ Q1 − Q1) − A1(˜ Q2 − Q2) D . In this notation (23) becomes [B2aj − A2bj](˜ Q1 − Q1) − [B1aj − A1bj](˜ Q2 − Q2) = [˜ qj − qj]D, which is precisely hj = ˜ hj, for j = 1, 2.

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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Reconstruction

The following reconstruction theorem also holds. Theorem Suppose we are given the data in Theorem 8. For each zj ∈ [0, 1] let (xki

j (mjv(ki)))i, j = 1, 2, be a subsequence of

(xk

j (mjv(k)))k, convergent to x and

gj(x) := 2

γaj(x) + bj(x) lim

i→∞ ¯

µki[¯ xk

j (mjv(ki)) − xk j (mjv(ki))].

Then gj(0) = 0 = gj(1) and gj′ γaj + bj = −¯ qj + Aaj + Bbj.

Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system

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A Short Bibiliography

P. A. BINDING, P. J. BROWNE, Asymptotics of eigencurves for second order ordinary differential equations, I
J. Differential Equations, 88 (1990), 30-45.
M. FAIERMAN, On the distribution of the eigenvalues of a two-parameter system of ordinary differential

equations of the second order, SIAM J. Math. Anal., 8 (1977), 854-870.

M. FAIERMAN, Distribution of eigenvalues of a two-parameter system of differential equations, Trans. Amer.
Math. Soc., 247 (1979), 45-86.

O.H. HALD, J.R. MCLAUGHLIN, Solutions of inverse nodal problems, Inverse Problems, 5 (1989), 307-347.

F. KLEIN, Über Körper, welche von confocalen Flächen zweiten Gradesbegrenzt sind, Math. Ann., 18

(1881), 410-427. J.R. MCLAUGHLIN, Inverse spectral theory using nodal points as data - a uniqueness result, J. Differential Equations, 73 (1988), 354–362. R.G.D. RICHARDSON, Theorems of oscillation for two linear differential equations of the second order with two parameters, Trans. Amer. Math. Soc., 13 (1912), 22-34. B.P. RYNNE, The asymptotic distribution of the eigenvalues of right definite multiparameter Sturm-Liouville systems, Proc. Edinburgh Math. Soc., 36 (1993), 35-47.

L. TURYN, Sturm-Liouville problems with several parameters, J. Differential Equations, 38 (1980), 239-259.

X.-F. YANG, A solution of the inverse nodal problem, Inverse Problems, 13 (1997), 203-213. Bruce A. Watson joint work with Paul Binding An inverse nodal problem for two-parameter Sturm-Liouville system