Topological index formula in physical waves: spectral flow, Chern index and topological contacts

We study a family of pseudodifferential operators (quantum Hamiltonians) on $L^{2}(\mathbb{R}^{n};\mathbb{C}^{d})$ whose spectrum exhibits two energy bands exchanging a finite number of eigenvalues. We show that this number coincides with the Chern index of a vector bundle associated to the principal symbol (the classical Hamiltonian). This result provides a simple yet illustrative instance of the Atiyah Singer index formula, with applications in areas such as molecular physics, plasma physics or geophysics. We also discuss the phenomenon of topological contact without exchange between energy bands, a feature that cannot be detected by the Chern index or K theory, but rather reflects subtle torsion effects in the homotopy groups of spheres.