On vanishing and localizing of transmission eigenfunctions near singular points: a numerical study
For researchers in spectral theory and inverse scattering, this provides novel numerical insights into the geometric structure of transmission eigenfunctions near singular points, though the study is limited to specific geometries and lacks rigorous proof.
This paper numerically investigates the behavior of interior transmission eigenfunctions near cusp singularities, finding that eigenfunctions vanish when the interior angle is less than π and localize when it is greater than π, with vanishing/blowup orders inversely proportional to the angle. These results are the first of their kind for transmission eigenfunctions.
This paper is concerned with the intrinsic geometric structure of interior transmission eigenfunctions arising in wave scattering theory. We numerically show that the aforementioned geometric structure can be much delicate and intriguing. The major findings can be roughly summarized as follows. If there is a cusp on the support of the underlying potential function, then the interior transmission eigenfunction vanishes near the cusp if its interior angle is less than $π$, whereas the interior transmission eigenfunction localizes near the cusp if its interior angle is bigger than $π$. Furthermore, we show that the vanishing and blowup orders are inversely proportional to the interior angle of the cusp: the sharper the angle, the higher the convergence order. Our results are first of its type in the spectral theory for transmission eigenvalue problems, and the existing studies in the literature concentrate more on the intrinsic properties of the transmission eigenvalues instead of the transmission eigenfunctions. Due to the limitedness of the computing resources, our study is by no means exclusive and complete. We consider our study only in a certain geometric setup including corner, curved corner and edge singularities. Nevertheless, we believe that similar results hold for more general cusp singularities and rigorous theoretical justifications are much desirable. Our study enriches the spectral theory for transmission eigenvalue problems. We also discuss its implication to inverse scattering theory.