  Bosonization provides a powerful analytical framework to deal with
one-dimensional strongly interacting fermion systems, which makes it a
cornerstone in quantum many-body theory. Yet, this success comes at the expense
of using effective infrared parameters, and restricting the description to low
energy states near the Fermi level. We propose a radical extension of the
bosonization technique that overcomes both limitations, allowing computations
with microscopic lattice Hamiltonians, from the Fermi level down to the bottom
of the band. The formalism rests on the simple idea of representing the fermion
kinetic term in the energy domain, after which it can be expressed in terms of
free bosonic degrees of freedom. As a result, one- and two-body fermionic
scattering processes generate anharmonic boson-boson interactions, even in the
forward channel. We show that up to moderate interaction strengths, these
nonlinearities can be treated analytically at all energy scales, using the
x-ray emission problem as a showcase. In the strong interaction regime, we
employ a systematic variational solution of the bosonic theory, and obtain
results that agree quantitatively with an exact diagonalization of the original
one-particle fermionic model. This provides a proof of the fully microscopic
character of bosonization on all energy scales for an arbitrary band structure.
Besides recovering the known x-ray edge singularity at the emission threshold,
we find strong signatures of correlations even at emission frequencies beyond
the band bottom.
