On the conicity of eigenvalues intersections for parameter-dependent self-adjoint operators

Motivated by recent controllability results for the bilinear Schrödinger equation based on the existence of conical intersections, in this paper we identify two physically interesting families of parameter-dependent Hamiltonians that admit residual and prevalent subfamilies for which all double eigenvalues are conical. In order to obtain such a result, we exploit a characterization of conical intersections in terms of a transversality condition which allows us to apply a suitable transversality theorem.