The rigged Hilbert space approach to the Lippmann–Schwinger equation: II. The analytic continuation of the Lippmann–Schwinger bras and kets

Abstract

Abstract. The analytic continuation of the Lippmann-Schwinger bras and kets is obtained and characterized. It is shown that the natural mathematical setting for the analytic continuation of the solutions of the Lippmann-Schwinger equation is the rigged Hilbert space rather than just the Hilbert space. It is also argued that this analytic continuation entails the imposition of a time asymmetric boundary condition upon the group time evolution, resulting into a semigroup time evolution. Physically, the semigroup time evolution is simply a (retarded or advanced) propagator.PACS numbers: 03.65.-w, 02.30.Hq IntroductionThis paper is devoted to construct and characterize the analytic continuation of the Lippmann-Schwinger bras and kets, as well as the analytic continuation of the "in" and "out" wave functions. This paper follows up on Ref.[1], where we obtained and characterized the solutions of the Lippmann-Schwinger equation associated with the energies of the physical spectrum. We showed in [1] that such solutions are accommodated by the rigged Hilbert space rather than by the Hilbert space alone. In this paper, we shall show that the analytic continuation of the Lippmann-Schwinger bras and kets is also accommodated by the rigged Hilbert space rather than by the Hilbert space alone.It was shown in Ref.[1] that the Lippmann-Schwinger bras and kets are distributions that act on a space of test functions Φ ≡ S(R + {a, b}). The space Φ arises from invariance under the action of the Hamiltonian and from the need to tame purely imaginary exponentials. These two requirements force the functions of Φ to have a polynomial falloff at infinity. The resulting Φ is a space of test functions of the Schwartz type. In this paper, it is shown that the analytic continuation of the LippmannSchwinger bras and kets are distributions that act on a space of test functions Φ exp . The space Φ exp arises from invariance under the action of the Hamiltonian and from the needThe RHS approach to the Lippmann-Schwinger equation II 2 to tame real exponentials. These two requirements force the elements of Φ exp to fall off at infinity faster than real exponentials. More precisely, we shall ask the elements of Φ exp to fall off faster than Gaussians. The resulting Φ exp is therefore of the ultradistribution type. We recall that an ultra-distribution is an infinitely differentiable test function that falls off at infinity faster than exponentials.In Ref.[1], we obtained the time evolution of wave functions and of the LippmannSchwinger bras and kets associated with real energies, and we saw that it is given by the standard quantum mechanical group time evolution. In this paper, we shall see that analytically continuing the time evolution of the wave functions results into a semigroup. We shall argue, although not fully prove, that analytically continuing the time evolution of Lippmann-Schwinger bras and kets also results into a semigroup.As in Ref.[1], we restrict ourselves to the spherical shell potentialfor zero angular momentum. Nevertheless, our...