In this thesis, we extend the theory of non-scattering energies on two fronts. First, we shall consider the discreteness of non-scattering energies corresponding to non-compactly supported potentials using the approach via transmission eigenvalues and fourth-order operators. The method requires the support of the potential to exhibit certain compact Sobolev embedding and to be contained in a half-space and the potential to have controlled polynomial or exponential decay at infinity. Also, in order to connect the non-scattering energies to the fourth-order operators, a generalization of the classical Rellich theorem to unbounded domains is required. This is of independent interest, and we obtain several such results, including a discrete analogue.
Our second contribution (joint work with L. Päivärinta and M. Salo) is extending a recent result on non-existence of non-scattering energies for potentials with rectangular corners to arbitrary corners of angle smaller than 180 degrees in two dimensions, and to prove in three dimensions that the set of strictly convex circular conical corners for which non-scattering energies might exist is at most countable.

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