  Operators on unbounded domains may acquire eigenvalues that are embedded in
the essential spectrum. Determining the fate of these embedded eigenvalues
under small perturbations of the underlying operator is a challenging task, and
the persistence properties of such eigenvalues is linked intimately to the
multiplicity of the essential spectrum. In this paper, we consider the planar
bilaplacian with potential and show that the set of potentials for which an
embedded eigenvalue persists is locally an infinite-dimensional manifold with
infinite codimension in an appropriate space of potentials.
