Transmission eigenvalues: promising for applications? (part I)

In the past two years I have been working on my licentiate thesis - now under evaluation by my supervisor - about transmission eigenvalues and non-scattering energies.

Let's try to explain in a simple way what the problem is. Imagine you have a medium (i.e. an object of study), of which you know the inner structure. Now imagine you can send some acoustic wave. With some computation, you can predict what the scattered wave will be like, that is you can predict how you wave will react to the medium's presence. To feel more comfortable, picture in your mind a real life situation: your medium is a rock in a pond and your wave is water waves.

On the left, the ingoing wave. On the right, the scattered wave.

Our inverse problem will be conceptually the following:

What can we guess about the inner structure of the medium, by observing how (some) acoustic waves interact with it?

In your simple example, imagine the rock is hidden from your sight and you have to give information about it (shape, etc.) just by observing the waves "deformed" by it. Not trivial, right?

As it usually happens with mathematicians, it all comes down to some partial differential equations, to be precise:

is the medium, aka "the ball" in the figure up here. is the index of refraction, aka a characterisation of the inner structure of . is what we are interested in: can we know something about it? is, say, the wave length of the wave you send in to monitorate . Finally, and are abstract functions closely related to ingoing and scattered waves. This system of equations model our situation (the interior transmission problem), under certain assumptions.

Such system can be seen as an eigenvalue problem, that is we may be interest in studying what frequencies 's serve our purpose to know since, in our model, is something we have control on. There are two interesting sets of frequencies:

non-scattering energies, i.e. 's such that the corresponding ingoing wave "ignores" . Nothing is observed.
transmission eigenvalues, i.e. 's such that the above system has a solution (with certain regularity requirements)

Clearly, transmission eigenvalue is a merely mathematical concept. What is most fascinating is that it has been proven that such eigenvalues carry important information about $D$. For instance if has an inner cavity, there is a relationship between the cavity size and the first transmission eigenvalue (*). What are they physically? How can we "measure" them?

I leave such questions for part II.

(*) see this paper by F. Cakoni, I. Harris and J. Sun.