Systematic upper and lower bounds for the real and imaginary parts of transition amplitudes.

Abstract

It is proved that the imaginary part of the diagonal matrix element 〈\ensuremath{\Psi}\ensuremath{\Vert}(E-H-i\ensuremath{\Gamma}${)}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\Vert}\ensuremath{\Psi}〉 of the Green's function is the maximum of a variational functional. This provides convenient lower bounds. Upper bounds to the imaginary part are deduced with the aid of Pad\'e approximants. Upper bounds of the real part of the transition amplitude are shown to follow from the unitarity relation. Finally, upper and lower bounds on nondiagonal matrix elements are derived and it is also shown that it is possible to obtain both upper and lower bounds on the real part of the transition amplitude. The negative consequence of employing these bounds is that we must deal with four-body operators.