Some generic properties of Schrödinger operators with radial potentials

We consider a class of Schrödinger operators on ℝ^Nwith radial potentials. Viewing them as self-adjoint operators on the space of radially symmetric functions in L²(ℝ^N), we show that the following properties are generic with respect to the potential: (P1) the eigenvalues below the essential spectrum are nonresonant (i.e., rationally independent) and so are the square roots of the moduli of these eigenvalues;(P2) the eigenfunctions corresponding to the eigenvalues below the essential spectrum are algebraically independent on any nonempty open set. The genericity means that in suitable topologies the potentials having the above properties form a residual set. As we explain, (P1), (P2) are prerequisites for some applications of KAM-type results to nonlinear elliptic equations. Similar properties also play a role in optimal control and other problems in linear and nonlinear partial differential equations.