Reciprocal Schrödinger Equation: Durations of Delay and of Final States Formation in Processes of Scattering

Abstract

The reciprocal Schrödinger equation $\partial S(\omega ,\mathbf{r}$ for $S$-matrix with temporal operator instead the Hamiltonian is established via the Legendre transformation of classical action function. Corresponding temporal functions are expressed via propagators of interacting fields. Their real parts ${\tau }_1$are equivalent to the Wigner-Smith delay durations at process of scattering and imaginary parts ${\tau }_2$ express the duration of final states formation (dressing). As an apparent example, they can be clearly interpreted in the oscillator model via polarization () and conductivity (${\tau }_2$) of medium. The $\tau$-functions are interconnected by the dispersion relations of Kramers-Krönig type. From them follows, in particular, that ${\tau }_2$ is twice bigger than the uncertainty value and thereby is measurable; it must be negative at some tunnel transitions and thus can explain the observed superluminal transfer of excitations at near field intervals (M.E.Perel'man. In: arXiv. physics/0309123). The covariant generalizations of reciprocal equation clarifies the adiabatic hypothesis of scattering theory as the requirement: at infinity future and elucidate the physical sense of some renormalization procedures.

28 pages