  We present the first calculation of the hadronic tensor on the lattice for
the nucleon. The hadronic tensor can be used to extract the structure functions
in deep inelastic scatterings and also provide information for the
neutrino-nucleon scattering which is crucial to the neutrino-nucleus scattering
experiments at low energies. The most challenging part in the calculation is to
solve an inverse problem. We have implemented and tested three algorithms using
mock data, showing that the Bayesian Reconstruction method has the best
resolution in extracting peak structures while the Backus-Gilbert and Maximum
Entropy methods are somewhat more stable for the flat spectral function.
Numerical results are presented for both the elastic case (clover fermions on
domain wall configuration with $m_\pi\sim$ 370 MeV and $a\sim$ 0.06 fm) and a
case (anisotropic clover lattice with $m_\pi\sim$ 380 MeV and $a_t\sim$ 0.035
fm) with large momentum transfer. For the former case, the reconstructed
Minkowski hadronic tensor gives precisely the vector charge which proves the
feasibility of the approach. While for the latter case, the nucleon resonances
and possibly shallow inelastic scattering contributions around $\nu=1$ GeV are
clearly observed but no information is obtained for higher excited states with
$\nu>2$ GeV. A check of the effective masses of $\rho$ meson with different
lattice setups indicates that, in order to reach higher energy transfers, using
lattices with smaller lattice spacings is essential.
