Abstract: MATH/CHEM/COMP 2002, Dubrovnik, June 24-29, 2002






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.