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

Abstract

The reciprocal Schrödinger equation $\partial S(ω,{\bf r}% )/i\partial ω=\hatτ(ω,{\bf r}) S(ω,{\bf r})$ 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 $τ_{1}$are equivalent to the Wigner-Smith delay durations at process of scattering and imaginary parts $τ_{2}$ express the duration of final states formation (dressing). As an apparent example, they can be clearly interpreted in the oscillator model via polarization ($% τ_{1}$) and conductivity ($τ_{2}$) of medium. The $τ$-functions are interconnected by the dispersion relations of Kramers-Krönig type. From them follows, in particular, that $τ_{2}$ 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: $% τ_{2}\to 0$ at infinity future and elucidate the physical sense of some renormalization procedures.