The recent development of a topological description of matter, especially for classical systems, is based on the formal analogy to quantum mechanics. The differential equation governing the classical system (Maxwell/Newton) with suitable boundary conditions is expressed by the generalized eigenvalue problem expanded by base functions. The gauge invariance introduced by a formal/fictitious gauge field is crucial. Then, natural constraints of the real QM are not trivial for the mapped fictitious quantum system/eigenvalue problem, for example, the positivity of the norm. Especially, the Hamiltonian/overlap matrices in most cases depend on the energy(eigenvalue). This implies that the eigenvalue problem is non-linear.
We then discuss this problem from a novel viewpoint, stressing the analytical description of the bulk-edge correspondence, which is fundamental for the experimental side. Look at our paper, “Bulk-edge correspondence for nonlinear eigenvalue problems”, by Takuma Isobe, Tsuneya Yoshida, and Yasuhiro Hatsugai, which has been published in Phys. Rev. Lett. on 19 March 2024. See also arXiv:2310.12577.