Absence of the analytic continuation of elastic transmission eigenfunctions at rectangular corners

We study time harmonic scattering problems in linear elasticity in $\mathbb{R}^{2}$. We show that certain penetrable scatterers with rectangular corners scatter every incident wave nontrivially. Even though these scatterers have interior transmission eigenvalues, the far field operator has a trivial kernel at every real frequency. Our approach relies on a special decomposition of the elastic Lam\'e operator and also provides an alternative idea for treating inverse elastic medium problems with a general polygonal support.