Evolution operator and Mpller operators

The time evolution of any system is determined (in the Schrbdinger picture of quantum mechanics) by the Schrodinger equation h = 1) [92]  

For conservative systems (which we shall always be considering) the Hamiltonian H is independent of t and the general solution of the Schrbdinger equation has the form  

It follows from a basic theorem on linear operators that, since H is selfadjoint, the evolution operator U t) is unitary (Jordan [93]). The evolution operator maps the state vector for time zero (that is, ip)) onto the corresponding vector for time t. 

The description of quantum scattering closely parallels the classical formalism. In lieu of the classical orbit x t) satisfying Newton s equation, we now have a state vector ipt) satisfying the time-dependent Schrodinger equation (3.1). I pt) is any vector in the appropriate Hilbert space H.We shall adopt the classical terminology and refer to the solution U(t) ip) as an orbits although of course it is no longer an orbit in real space R. Every orbit U (t) ip) can be uniquely identified by the fixed vector ip), which is just the state vector at the instant =0. 

We are considering a single spinless particle in a fixed local potential. Let us suppose that the orbit U(t) ip) describes the evolution of some scattering experiment. This means that when followed back to a time well before the collision U (t) ip) represents a wave packet that is localized far away from the scattering center and, therefore, behaves like a free wave packet. Now, the motion of a free particle is given by the free evolution operator 

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