Evaluation of $2νββ$ decay nuclear matrix elements using Lanczos techniques preserving Gamow-Teller strengths from both parent and daughter nuclei

We present two efficient shell-model approaches for calculating nuclear matrix elements (NMEs) for two-neutrino double-beta ($2νββ$) decay and two-neutrino double-electron capture ($2ν$ECEC), based on the Lanczos strength-function and block-Lanczos techniques. The dual-sided Lanczos strength-function (DLSF) and block-Lanczos strength-function (BLSF) methods avoid the explicit calculation of a complete set of exact intermediate 1⁺ eigenstates and yield fully converged NMEs with a relatively small number of iterations starting from Gamow--Teller (GT) doorway states. Both methods simultaneously preserve the total GT strengths associated with the initial and final nuclei. We benchmark the methods for ⁴⁸Ca in the $fp$ shell and apply them to candidates in different mass regions, including ⁵⁸Ni, ⁷⁶Ge, and ¹³⁰Te. For ⁴⁸Ca, a calculation allowing cross-shell excitations of up to 2ω in the $sdpf$ model space yields a $2νββ$ NME closer to the experimental estimate than the result obtained in the $fp$ shell. For ⁵⁸Ni, we present, to our knowledge, the first shell-model calculation of the 2νβ⁺EC/$2ν$ECEC NME. Comparison with the current experimental upper limit on the NME indicates the need for a more sensitive measurement of the 2νβ⁺EC half-life. As an independent validation, the energy-denominator-free sums obtained with the proposed methods are compared with the corresponding double-GT matrix elements calculated directly between the initial and final ground states. Their agreement demonstrates that the methods correctly recover the complete energy-denominator-free intermediate-state sums.