An Inverse Spectral Problem for Inhomogeneous Media Sam Cogar - PowerPoint PPT Presentation

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 ∈ L 2 (0 , 1) and h , H ∈ R with h 2 + H 2 > 0, find the eigenvalues λ and eigenfunctions u � = 0 such that − u ′′ ( x ) + q ( x ) u ( x ) = λ u ( x ) , 0 ≤ x ≤ 1 , u (0) = 0 and hu ′ (1) + Hu (1) = 0 . In this example, there exist countable many eigenvalues { λ n } . Sam Cogar (University of Delaware) An Inverse Spectral Problem October 5, 2016 3 / 17

Inverse Eigenvalue Problems The Inverse Problem Given the eigenvalues of some PDE eigenvalue problem (and possibly other information), determine some coefficient(s) in the problem. An Example: Inverse Sturm-Liouville Problems Given h , H ∈ R with h 2 + H 2 > 0 and the eigenvalues { λ n } of the problem − u ′′ ( x ) + q ( x ) u ( x ) = λ u ( 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 . Sam Cogar (University of Delaware) An Inverse Spectral Problem October 5, 2016 4 / 17

Inverse Eigenvalue Problems A Negative Result Let λ be an eigenvalue and u a corresponding eigenfunction of − u ′′ ( x ) + q ( x ) u ( x ) = λ u ( 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 ) = λ v ( x ) , 0 < x < 1 , v (0) = 0 , v (1) = 0 , where ˜ q ( x ) = q (1 − x ). [Kirsch, 2011] Sam Cogar (University of Delaware) An Inverse Spectral Problem October 5, 2016 5 / 17

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 . Sam Cogar (University of Delaware) An Inverse Spectral Problem October 5, 2016 6 / 17

Scattering in Inhomogeneous Media The Direct Problem Given the refractive index n ( x ) and a unit vector d ∈ R 3 , find the total field u such that ∆ u + k 2 n ( x ) u = 0 in R 3 u = e ikx · d + u s � ∂ u s � ∂ r − iku s lim = 0 . r →∞ r The scattered field u s has the asymptotic behavior � 1 u s ( x ) = e ik | x | � | x | u ∞ (ˆ x , d ) + O as | x | → ∞ , | x | 2 x where ˆ x = | x | and u ∞ (ˆ x , d ) is the far field pattern . Sam Cogar (University of Delaware) An Inverse Spectral Problem October 5, 2016 8 / 17

Scattering in Inhomogeneous Media The Inverse Problem x , d ∈ R 3 , Given the far field pattern u ∞ (ˆ x , d ) for all unit vectors ˆ 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 ∈ R 3 | n ( x ) � = 1 } . Sam Cogar (University of Delaware) An Inverse Spectral Problem October 5, 2016 9 / 17

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 + k 2 n ( x ) w = 0 , ∆ v + k 2 v = 0 in D ∂ w ∂ν = ∂ v w = 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 ). Sam Cogar (University of Delaware) An Inverse Spectral Problem October 5, 2016 10 / 17

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 R 3 - { k j } 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 { k j } uniquely determines n ( r ). What about if n ( r ) > 1 for 0 ≤ r < 1? Sam Cogar (University of Delaware) An Inverse Spectral Problem October 5, 2016 11 / 17

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 + k 2 n ( x ) w = 0 , ∆ v + k 2 η 2 v = 0 in D ∂ w ∂ν = ∂ v w = v , ∂ν on ∂ D . Physically speaking, modified transmission eigenvalues correspond to scattering identical to a medium of constant refractive index. Sam Cogar (University of Delaware) An Inverse Spectral Problem October 5, 2016 12 / 17

Scattering in Inhomogeneous Media An Inverse Spectral Problem Assume that: - n ( x ) = n ( r ) is radially symmetric - D is the unit ball in R 3 - { k j } are the transmission eigenvalues (including multiplicities) with radially symmetric eigenfunctions - { ˜ k j } 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 ) < η 2 for 0 ≤ r < 1, then knowledge of { k j } and { ˜ k j } uniquely determines n ( r ). Sam Cogar (University of Delaware) An Inverse Spectral Problem October 5, 2016 13 / 17

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 { z j } j ∈ N are its zeros (other than the origin) repeated according to their multiplicities, then ∞ � � 1 − z 2 � f ( z ) = cz m , z 2 j j =1 where m is the multiplicity of the zero at the origin and c is a constant. Sam Cogar (University of Delaware) An Inverse Spectral Problem October 5, 2016 14 / 17

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. Sam Cogar (University of Delaware) An Inverse Spectral Problem October 5, 2016 15 / 17

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. Sam Cogar (University of Delaware) An Inverse Spectral Problem October 5, 2016 16 / 17

Scattering in Inhomogeneous Media Questions? Sam Cogar (University of Delaware) An Inverse Spectral Problem October 5, 2016 17 / 17