Efficient method for evaluating energy-dependent sum rules

Energy-dependent sum rules are useful tools in many fields of physics. In nuclear physics, they typically involve an integration of the response function over the nuclear spectrum with a weight function composed of integer powers of the energy. More complicated weight functions are also encountered, e.g., in nuclear polarization corrections of atomic spectra. Using the Lorentz integral transform method and the Lanczos algorithm, we derive a computationally efficient technique for evaluating such sum rules that avoids the explicit calculation of both the continuum states and the response function itself. Our numerical results for electric dipole sum rules of the ${}^4\mathrm{He}$ nucleus with various energy-dependent weights show rapid convergence with respect to the number of Lanczos steps. This demonstrates the usefulness of the method in a variety of electroweak reactions.

Figure 1

The Gaussian-basis-related function $h_i(\sigma ,\Gamma )$, Eq. (19), plotted as a function of the argument $(\sigma -{\omega }_i)/{\beta }_i$ for various values of $\Gamma /{\beta }_i$, rescaled by division with its peak value at $\sigma ={\omega }_i$.

Figure 2

Convergence of various energy-dependent sum rules with increasing number of Lanczos steps. The calculations are done using the unretarded dipole response of ${}^4\mathrm{He}$. The sum rules are normalized to their converged values.