Exact mathematical treatment of the modifications of finite-dimensional quantum systems

Low rank modification (LRM) is a new mathematical formalism by which one can express eigenvalues and eigenstates of the modified system B in terms of the eigenvalues and eigenstates of the original system A. In this respect, LRM is similar to a standard perturbation expansion, which also expresses eigenvalues and eigenstates of the perturbed system B in terms of the eigenvalues and eigenstates of the unperturbed system A. However, unlike perturbation expansion, LRM produces correct results however strong the “perturbation” of the original system A. LRM is here applied to finite n‐dimensional systems A and B that are described by generalized n × n eigenvalue equations. In the LRM approach, modified system B is described by a ρ × ρ matrix equation, where ρ is the dimension of the space affected by the “modification” of the original system A. In mathematical terms, ρ is the rank of the operators that describe this modification. In many important cases, ρ