Abstract: MATH/CHEM/COMP 2002, Dubrovnik, June 24-29, 2002
INTERACTION OF AN EIGENSTATE WITH THE
ONE-PARAMETER EIGENVALUE BAND
Tomislav P. zivkovic
Rudjer Boskovic Institute, POB 180, HR-1000 Zagreb, Croatia
Interaction of a quantum system containing a single state with a known infinite dimensional quantum system containing an eigenvalue band is considered. A new approach for the treatment of the combined system is developed. This system contains embedded eigenstates with continuous eigenvalues , and in addition it may contain isolated eigenstates with discrete eigenvalues . Exact expressions for the solution of the combined system are derived. In particular, due to the interaction with the system , eigenvalue of the state shifts and in addition if 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 that is initially prepared in the state 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 with the system , however strong. It also presents a relatively good approximation for the interaction of a nondegenerate eigenstate of an arbitrary system with an infinite system containing a single eigenvalue band, provided this eigenstate is relatively well separated from other eigenstates of and provided the interaction between the systems and is not excessively strong.