ANNALS OF PHYSICS: 51, 187-200 (1969) Explicit Solution of the Continuous Baker-Campbell-Hausdorff Problem and a New Expression for the Phase Operator I. BIALYNICKI-BIRULA, B. MIELNIK, AND J. PLEBA~~SKI Institute of Theoretical Physics, Warsaw University, Warsaw, Poland An explicit formula for an arbitrary function of the evolution operator is derived. With its use, the continuous analog of the Baker-Campbell-Hausdorff problem is solved. The application of this result to the quantum theory of scattering leads to a new closed expression for the phase shifts in every order of perturbation theory. I. INTRODUCTION In many branches of physics and mathematics we are led to study the evolution equation (1.1) The purpose of this paper is the derivation of a new representation for an arbitrary function of E, the evolution 0perator.l Using this representation for the function Sz = In E, we obtain the solution of the continuous analog of the Baker-Campbell-Hausdorff problem (I), (2) in a closed form. We discuss also briefly in this paper one of the most interesting applications of our result, namely, to the scattering theory in quantum mechanics, and we derive an explicit formula for the phase shifts in every order of perturbation theory. The evolution equation (1.1) and the initial condition for the evolution operator, J%l 9 44 = 1, (1.2) are equivalent to the following integral equation 1 We shall call traditionally all objects like A(t), E(r, &,) etc. the operators even though they need not be defined as operators in a vector space. All we shall need is that they form an asso- ciative algebra. 187