Stability Conditions in Quantum System

A systematic method is presented for studying the stability of the solution of a quantum system with both zero temperature and non-zero temperature. It employs the generalized action functional and its first and second derivatives. The zero eigenvalue and corresponding eigenvector of the matrix constructed by the second derivative of the action functional determine the eigenmode of the spectrum. It gives the generalized on-shell condition applicable to the space-time translation non-invariant case also. The method provides at the same time a general formalism of deriving exact bound state equation.