Electronic Final States in Nuclear $\beta$ Decay: A Sudden-Approximation Framework

Electronic final states generated by sudden changes of the Hamiltonian are studied here, with emphasis on nuclear charge variation in $\beta$ decay. A $\lambda$-parametrized family $\widehat{H}(\lambda )$ that continuously connects the initial and final Hamiltonians, so that the electronic response can be represented as a continuous deformation in Hilbert space, is introduced. Within the sudden approximation, transition amplitudes are written as overlaps between eigenstates of distinct Hamiltonians. To relate non-orthogonal one-electron basis sets in a stable way, the paper uses a practical transport scheme based on overlap metrics and truncated singular value decomposition (SVD). This mapping is interpreted as a discrete counterpart of continuous transport along the $\lambda$ path. The formalism is first developed for the one-electron case, where analytic structure and selection rules are made explicit, and then generalized to many-electron systems via nonorthogonal determinant overlap expressions. The resulting formulation gives transition probabilities in bound and continuum channels in a way that is both numerically stable and easy to interpret.