Interaction of a finite quantum system with an infinite quantum system that contains a single one-parameter eigenvalue band

Abstract

Interaction of a finite quantum system $${\textsf{\textbf{S}}_\rho^a}$$ that contains ρ eigenvalues and eigenstates with an infinite quantum system $${\textsf{\textbf{S}}_\infty^{\rm b}}$$ that contains a single one-parameter eigenvalue band is considered. A new approach for the treatment of the combined system $${\textsf{\textbf{S}}_\infty \equiv \textsf{\textbf{S}}_\rho^a \oplus \textsf{\textbf{S}}_\infty^b}$$ is developed. This system contains embedded eigenstates $${\left| {\psi (\epsilon)} \right\rangle}$$ with continuous eigenvalues $${\varepsilon}$$ , and, in addition, it may contain isolated eigenstates $${\left| {\psi_s } \right\rangle}$$ with discrete eigenvalues $${\varepsilon_s}$$ . Two ρ × ρ eigenvalue equations, a generic eigenvalue equation and a fractional shift eigenvalue equation are derived. It is shown that all properties of the system $${\textsf{\textbf{S}}_\rho^a}$$ that interacts with the system $${\textsf{\textbf{S}}_\infty^{\rm b}}$$ can be expressed in terms of the solutions to those two equations. The suggested method produces correct results, however strong the interaction between quantum systems $${\textsf{\textbf{S}}_\rho^a}$$ and $${\textsf{\textbf{S}}_\infty^{\rm b}}$$ . In the case of the weak interaction this method reproduces results that are usually obtained within the formalism of the perturbation expansion approach. However, if the interaction is strong one may encounter new phenomena with much more complex behavior. This is also the region where standard perturbation expansion fails. The method is illustrated with an example of a two-dimensional system $${\textsf{\textbf{S}}_2^a}$$ that interacts with the infinite system $${\textsf{\textbf{S}}_\infty^{\rm b}}$$ that contains a single one-parameter eigenvalue band. It is shown that all relevant completeness relations are satisfied, however strong the interaction between those two systems. This provides a strong verification of the suggested method.