Communicated by the Editors

We consider the general Cauchy problem with initial data in a Hilbert space and with a formal dissipative linear generator. A complete parametrization is known of the (abstract) boundary conditions which make this problem well set. We exhibit a distinguished subset a ~ of the set I of boundary conditions and demonstrate explicitly that the evolution associated with each B in B can be represented as a (time independent) average over the evolutions associated with B ’ in L%x, Applications are discussed to Schrijdinger equations in bounded regions or with singular potentials. Consider the Cauchy problem 1