Long-time dynamics via direct summation of infinite continued fractions

Mori theory leads to in(finite) continued fractions [I(F)CF’s] which upon inverse Laplace transformation (ILT) give the dynamical correlations in Hamiltonian systems. We propose a direct summation method to evaluate these ICF’s, e.g., 1/[z+${\mathrm{\Delta }}_1$/(z+... ∞)], by replacing them with FCF’s with poles L=${10}^{\mathrm{\zeta }}$, 2≤ζ≤5, for ${\mathrm{\Delta }}_{\mathrm{\mu }}$=${\mathrm{\mu }}^{\mathrm{\phi }}$, φ<2. Long-time dynamics is obtained upon an ILT of the ICF for 0≤t≤τ with τ=f(φ,ζ) being large. Our studies on dynamical correlations for boundary spins in S=1/2 XY chains agree very well with a recent exact solution for these correlations.