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An Inverse Spectral Problem for Inhomogeneous Media

Sam Cogar (Joint work with David Colton and YJ Leung)

Department of Mathematical Sciences University of Delaware

October 5, 2016 Research supported by AFOSR and ARL

Sam Cogar (University of Delaware) An Inverse Spectral Problem October 5, 2016 1 / 17

Inverse Eigenvalue Problems

The Direct Problem

Given some PDE eigenvalue problem, determine its eigenvalues and corresponding eigenfunctions. An Example: Sturm-Liouville Problems Given a function q ∈ L2(0, 1) and h, H ∈ R with h2 + H2 > 0, find the eigenvalues λ and eigenfunctions u = 0 such that −u′′(x) + q(x)u(x) = λu(x), 0 ≤ x ≤ 1, u(0) = 0 and hu′(1) + Hu(1) = 0. In this example, there exist countable many eigenvalues {λn}.

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Inverse Eigenvalue Problems

The Inverse Problem

Given the eigenvalues of some PDE eigenvalue problem (and possibly

• ther information), determine some coefficient(s) in the problem.

An Example: Inverse Sturm-Liouville Problems Given h, H ∈ R with h2 + H2 > 0 and the eigenvalues {λn} of the problem −u′′(x) + q(x)u(x) = λu(x), 0 < x < 1, u(0) = 0 and hu′(1) + Hu(1) = 0, determine the coefficient function q. The bad news: in general, these eigenvalues do not uniquely determine q.

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Inverse Eigenvalue Problems

A Negative Result

Let λ be an eigenvalue and u a corresponding eigenfunction of −u′′(x) + q(x)u(x) = λu(x), 0 < x < 1, u(0) = 0, u(1) = 0. Then λ is also an eigenvalue with corresponding eigenfunction v(x) = u(1 − x) of −v′′(x) + ˜ q(x)v(x) = λv(x), 0 < x < 1, v(0) = 0, v(1) = 0, where ˜ q(x) = q(1 − x). [Kirsch, 2011]

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Inverse Eigenvalue Problems

The Inverse Problem

In general, we must know additional information in order to determine q, such as a second set of eigenvalues or some properties of q.

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Scattering in Inhomogeneous Media

The Direct Problem

Given the refractive index n(x) and a unit vector d ∈ R3, find the total field u such that ∆u + k2n(x)u = 0 in R3 u = eikx·d + us lim

r→∞ r

∂us ∂r − ikus

• = 0.

The scattered field us has the asymptotic behavior us(x) = eik|x| |x| u∞(ˆ x, d) + O 1 |x|2

• as |x| → ∞,

where ˆ x =

x |x| and u∞(ˆ

x, d) is the far field pattern.

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Scattering in Inhomogeneous Media

The Inverse Problem

Given the far field pattern u∞(ˆ x, d) for all unit vectors ˆ x, d ∈ R3, determine the refractive index n(x). This problem has a unique solution [Theorem 10.5, Colton-Kress], but it is ill-posed. The linear sampling method gives an approximation of D = {x ∈ R3|n(x) = 1}.

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Scattering in Inhomogeneous Media

Transmission Eigenvalues

Definition We say that k > 0 is a transmission eigenvalue if there exists a nontrivial pair w, v for which ∆w + k2n(x)w = 0, ∆v + k2v = 0 in D w = v, ∂w ∂ν = ∂v ∂ν on ∂D. Physically speaking, transmission eigenvalues correspond to the non-scattering of special incident fields. Transmission eigenvalues may be computed from far field patterns, and they carry information about n(x).

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Scattering in Inhomogeneous Media

An Inverse Spectral Problem

Question: Do the transmission eigenvalues uniquely determine n(x)? Assume that:

• n(x) = n(r) is radially symmetric
• D is the unit ball in R3
• {kj} are the transmission eigenvalues (including multiplicities)

with radially symmetric eigenfunctions Theorem (Colton-Leung, 2013) If n ∈ C 3[0, 1], n(1) = 1, n′(1) = 0, and 0 < n(r) < 1 for 0 ≤ r < 1, then knowledge of {kj} uniquely determines n(r). What about if n(r) > 1 for 0 ≤ r < 1?

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Scattering in Inhomogeneous Media

Modified Transmission Eigenvalues

As in the case of inverse Sturm-Liouville problems, we need a second set of eigenvalues. Definition Let η > 0 with η = 1. We say that k > 0 is a modified transmission eigenvalue if there exists a nontrivial pair w, v for which ∆w + k2n(x)w = 0, ∆v + k2η2v = 0 in D w = v, ∂w ∂ν = ∂v ∂ν on ∂D. Physically speaking, modified transmission eigenvalues correspond to scattering identical to a medium of constant refractive index.

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Scattering in Inhomogeneous Media

An Inverse Spectral Problem

Assume that:

• n(x) = n(r) is radially symmetric
• D is the unit ball in R3
• {kj} are the transmission eigenvalues (including multiplicities)

with radially symmetric eigenfunctions

• {˜

kj} are the modified transmission eigenvalues (including multiplicities) for η = 1 with radially symmetric eigenfunctions A New Result If n ∈ C 3[0, 1], n(1) = 1, n′(1) = 0, and 1 < n(r) < η2 for 0 ≤ r < 1, then knowledge of {kj} and {˜ kj} uniquely determines n(r).

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Scattering in Inhomogeneous Media

Some Tools for the Proof

• Transformation operators and the Liouville transformation
• Asymptotic behavior of solutions to ODEs
• Completeness results for trigonometric functions

Hadamard Factorization Theorem (Special Case) If f : C → C is an even entire function of exponential type of order one and {zj}j∈N are its zeros (other than the origin) repeated according to their multiplicities, then f (z) = czm

∞

• j=1
• 1 − z2

z2

j

• ,

where m is the multiplicity of the zero at the origin and c is a constant.

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Scattering in Inhomogeneous Media

Future Work

• 1. Remove the condition that n(1) = 1 and n′(1) = 0.
• 2. Investigate the existence and location of modified transmission

eigenvalues in the complex plane.

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Scattering in Inhomogeneous Media

References

• 1. Cakoni F, Colton D (2014) A Qualitative Approach to Inverse

Scattering Theory. Springer, New York.

• 2. Colton D, Kress R (2013) Inverse Acoustic and Electromagnetic

Scattering Theory, 3rd edn. Springer, New York.

• 3. Kirsch A (2011) An Introduction to the Mathematical Theory of

Inverse Problems, 2nd edn. Springer, New York.

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Scattering in Inhomogeneous Media

Questions?

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