Interaction of a Single State with a Known Infinite System Containing One-Parameter Eigenvalue Band

Interaction of a quantum system S ₁ ^a containing a single state |> with a known infinite-dimensional quantum system S ^b containing an eigenvalue band _a , _b ] is considered. A new approach for the treatment of the combined system S =S ₁ ^a S ^b is developed. This system contains embedded eigenstates |()> with continuous eigenvalues _a , _b ], and, in addition, it may contain isolated eigenstates | _I > with discrete eigenvalues _{I a} , _b ]. Exact expressions for the solution of the combined system are derived. In particular, due to the interaction with the system S ^b , eigenvalue E of the state |> shifts and, in addition, if E _a , _b ] this shifted eigenvalue broadens. Exact expressions for the eigenvalue shift and for the eigenvalue distribution of the state |> are derived. In the case of the weak coupling this eigenvalue distribution reduces to the standard resonance curve. Also, exact expressions for the time evolution of the state |(t)> that is initially prepared in the state |(0)>|> are obtained. Here again in the case of the weak coupling this time evolution reduces to the familiar exponential decay. The suggested method is exact and it applies to each coupling of the system S ₁ ^a with the system S ^b , however strong. It also presents a relatively good approximation for the interaction of a nondegenerate eigenstate | _s > of an arbitrary system S ^a with an infinite system S ^b containing a single eigenvalue band, provided this eigenstate is relatively well separated from other eigenstates of S ^a and provided the interaction between the systems S ^a and S ^b is not excessively strong.