Nondegeneracy conjecture for critical points of the Bloch variety

Prove that every critical point of the coordinate function λ on the complex Bloch variety defined by the dispersion relation det(Â(z) − λ I) = 0 for a periodic graph operator is nondegenerate.

Background

Beyond behavior at spectral edges, the paper considers all critical points of the energy coordinate λ on the complex Bloch variety of a periodic graph operator. A stronger assertion than the spectral edges conjecture is that every such critical point is nondegenerate, which would have broad implications for spectral analysis and related physical phenomena.

The authors present a dichotomy result showing that, for a given graph, generic parameter choices either yield all critical points nondegenerate or all degenerate, and they discuss methods to compute and bound the total number of critical points. Nonetheless, the global assertion that every critical point is nondegenerate remains framed as a conjecture.