On the interaction of two finite-dimensional quantum systems

Interaction of quantum system S-a described by the generalised rho x rho eigenvalue equation A\ Theta s] = EsSa\ Theta(s)] (s = 1,...,rho) with quantum system S-b described by the generalised n x n eigenvalue equation B\Phi(i)] = lambda(i)S(b)\Phi(i)] (i = 1,..., n) is considered. With the system S-a is associated rho-dimensional space X-rho(a) and with the system S-b is associated an n-dimensional space X-n(b) that is orthogonal to X-rho(a). Combined system S is described by the generalised (rho + n) x (rho + n) eigenvalue equation [A + B + V]\Psi(k) = epsilon(k)[S-a + S-b + P]\Psi(k)] (k = 1,..., n + rho) where operators V and P represent interaction between those two systems. All operators are Hermitian, while operators S-a, S-b and S = S-a + S-b+ P are, in addition, positive definite. It is shown that each eigenvalue epsilon(k) is not an element of {lambda(i)} of the combined system is the eigenvalue of the rho x rho eigenvalue equation [Omega(epsilon(k)) + A]\Psi(k)(a)] = epsilon(k)S(a)\Psi(k)(a)]. Operator Omega(epsilon) in this equation is expressed in terms of the eigenvalues lambda(i) of the system S-b and in terms of matrix elements [chi(s)\V\Phi(i)] and [chi(s)|P|Phi(i)] where vectors \chi(s)] form a base in X-rho(a). Eigenstate \Psi(k)(a)] of this equation is the projection of the eigenstate \Psi(k)] of the combined system on the space X-rho(a). Projection \Psi(k)(b) of \Psi(k)] on the space X-n(b) is given by \Psi(k)(b)] = (epsilon(k)S(b) - B)(-1) (V - epsilon(k)P)\Psi(k)(a)] where (epsilon(k)S(b) - B)(-1) is inverse of (epsilon(k)S(b) - B) in X-n(b). Hence, if the solution to the system S-b is known, one can obtain all eigenvalues epsilon(k) is not an element of {lambda(i)} and all the corresponding eigenstates \Psi(k)] of the combined system as a solution of the above rho x rho eigenvalue equation that refers to the system S-a alone. Slightly more complicated expressions are obtained for the eigenvalues epsilon(k) is not an element of {lambda(i)} and the corresponding eigenstates, provided such eigenvalues and eigenstates exist.