Hamiltonian reconstruction: The correlation matrix and incomplete operator bases

We explore the robustness of the correlation matrix Hamiltonian reconstruction technique with respect to the choice of operator basis, studying the effects of bases that are undercomplete and overcomplete---too few or too many operators, respectively. An approximation scheme for reconstruction using an undercomplete basis is proposed and tested numerically on select models. We discuss the confounding effects of conserved quantities and symmetries on reconstruction attempts. We apply these considerations to a variety of one-dimensional systems at zero and nonzero temperature.