In this talk, we are concerned with the properties of the spectrum of some families of parameter-dependent self-adjoint operators. More precisely, the property we are interested in the following question: how frequent are conical intersections between eigenvalues? This question is motivated by a series of control protocols of the bilinear Schrödinger equation, based on the existence of conical intersections.

In this talk, 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.

This is a joint work with Paolo Mason.
On the conicity of eigenvalues intersections for parameter-dependent self-adjoint operators
J. Math. Phys. 61, 053503 (2020); doi: 10.1063/1.5115576