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In many applied problems one seeks to identify and count the critical pointsof a particular eigenvalue of a smooth parametric family of self-adjointmatrices, with the parameter space often being known and simple, such as atorus. Among particular settings where such a question arises are theFloquet--Bloch decomposition of periodic Schroedinger operators, topology ofpotential energy surfaces in quantum chemistry, spectral optimization problemssuch as minimal spectral partitions of manifolds, as well as nodal statisticsof graph eigenfunctions. In contrast to the classical Morse theory dealing withsmooth functions, the eigenvalues of families of self-adjoint matrices are notsmooth at the points corresponding to repeated eigenvalues (called, dependingon the application and on the dimension of the parameter space, thediabolical/Dirac/Weyl points or the conical intersections).  This work develops a procedure for associating a Morse polynomial to a pointof eigenvalue multiplicity; it utilizes the assumptions of smoothness andself-adjointness of the family to provide concrete answers. In particular, wedefine the notions of non-degenerate topologically critical point andgeneralized Morse family, establish that generalized Morse families are genericin an appropriate sense, establish a differential first-order conditions forcriticality, as well as compute the local contribution of a topologicallycritical point to the Morse polynomial. Remarkably, the non-smooth contributionto the Morse polynomial turns out to be universal: it depends only on the sizeof the eigenvalue multiplicity and the relative position of the eigenvalue ofinterest and not on the particulars of the operator family; it is expressed interms of the homologies of Grassmannians.

Morse inequalities for ordered eigenvalues of generic self-adjoint families