Hamiltonian as a Hessian on the Hibert space, eigenvectors as critical points, and their relation to topological invariants in the variation method

Abstract

If a quantum vector varies in the Hilbert space, as trial functions do in the variational method, a vector field gets defined whose critical points are the eigenvectors of the Hamiltonian. The numbers of each type of critical point (minima, maxima, saddle points of various “indices”) are related to the topology of the compact variety, the closed multidimensional surface on which the trial vectors wander when they are restricted to unit normalization. The “global” results from that approach are compared with those of the “local” theory in which the type of each critical point is obtained from the Hessian on the Hilbert space whose eigenvalues are derived in terms of those of the Hamiltonian involved in the vector field. In a configuration-interaction (CI) problem for example, the type of saddle point each “excited state” represents is determined.