Semiclassical representation of width-weighted spectra

We consider a system having decaying states with complex energies E-j - i Gamma(j)/2, and we define the "width-weighted spectrum" as D Gamma(E) = Sigma(j) Gamma(j)delta (E - E-j) We derive a semiclassical formula for this width-weighted spectrum, a formula which is analogous to the periodic-orbit representation of the density of states. The formula applies if classical motion is regular and if decay occurs by tunneling through a barrier. The semiclassical formula involves not periodic orbits or closed orbits, but action integrals associated with irreducible loops on the "extremal torus," on which the particle climbs up the barrier and hangs at the top. Calculations confirm the validity of the formula. [S0031-9007(98)07690-X].