On the determination of Dirichlet or transmission eigenvalues from far field data

We show that the Herglotz wave function with kernel the Tikhonov regularized solution of the far field equation becomes unbounded as the regularization parameter tends to zero iff the wavenumber k belongs to a discrete set of values. When the scatterer is such that the total field vanishes on the boundary, these values correspond to the square root of Dirichlet eigenvalues for -Δ. When the scatterer is a nonabsorbing inhomogeneous medium these values correspond to so-called transmission eigenvalues.