  In many applied problems one seeks to identify and count the critical points
of a particular eigenvalue of a smooth parametric family of self-adjoint
matrices, with the parameter space often being known and simple, such as a
torus. Among particular settings where such a question arises are the
Floquet--Bloch decomposition of periodic Schr\"odinger operators, topology of
potential energy surfaces in quantum chemistry, spectral optimization problems
such as minimal spectral partitions of manifolds, as well as nodal statistics
of graph eigenfunctions. In contrast to the classical Morse theory dealing with
smooth functions, the eigenvalues of families of self-adjoint matrices are not
smooth at the points corresponding to repeated eigenvalues (called, depending
on the application and on the dimension of the parameter space, the
diabolical/Dirac/Weyl points or the conical intersections).
  This work develops a procedure for associating a Morse polynomial to a point
of eigenvalue multiplicity; it utilizes the assumptions of smoothness and
self-adjointness of the family to provide concrete answers. In particular, we
define the notions of non-degenerate topologically critical point and
generalized Morse family, establish that generalized Morse families are generic
in an appropriate sense, establish a differential first-order conditions for
criticality, as well as compute the local contribution of a topologically
critical point to the Morse polynomial. Remarkably, the non-smooth contribution
to the Morse polynomial turns out to depend only on the size of the eigenvalue
multiplicity and the relative position of the eigenvalue of interest and not on
the particulars of the operator family; it is expressed in terms of the
homologies of Grassmannians.
