Reciprocal Schr\"{o}dinger Equation: Durations of Delay and of Final States Formation in Processes of Scattering

Abstract

The reciprocal Schr\"{o}dinger equation $\partial S(\omega ,{\bf r}% )/i\partial \omega =\hat{\tau}(\omega ,{\bf r}) S(\omega ,{\bf r})$ 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 $\tau_{1}$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 ($% \tau_{1}$) and conductivity ($\tau_{2}$) of medium. The $\tau $-functions are interconnected by the dispersion relations of Kramers-Kr\"{o}nig 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: $% \tau_{2}\to 0$ at infinity future and elucidate the physical sense of some renormalization procedures.