Topology of the Ritz energy landscape

Motivated by questions from quantum chemistry and spectral optimization, this talk explores the topology of the Ritz energy landscape. Ritz values are the eigenvalues of a larger matrix (or operator) B compressed to a smaller trial subspace S. We can view the k-th Ritz value as a real-valued function (“Ritz energy landscape”) on the manifold of all possible
s-dimensional trial subspaces, the Grassmannian Gr(n,s).

We show that the k-th Ritz value (for any k) is a perfect Morse function, once the definition of “perfection” is suitably adjusted. A Morse function is called perfect if it describes the topology of its domain in the most efficient way possible, meaning the number of its critical points of each type exactly matches the corresponding Betti number of the space. While the k-th eigenvalue is Lipschitz rather than smooth and while its critical points are not isolated — and not even Morse-Bott — its critical point count is nevertheless well-defined and reflects the topology of the Grassmannian in a minimal, perfect way.

The proof proceeds by introducing a suitable perturbation which ensures that points of non-smoothness are not critical (by a theorem of Zelenko and the presenter) and that the critical points that remain are isolated. We also show that the notion of perfection we introduce (“homological perfection”) is closed under such perturbations.

Based on joint work with Mark Goresky (IAS).