Energy Range of State Conservation for Coulomb Interaction Effects in $La_{1.28}Sr_{1.72}Mn_2O_7$

Tunneling data on $La_{1.28}Sr_{1.72}Mn_2O_7$ crystals confirm Coulomb interaction effects through the $\sqrt{E}$ dependence of the density of states (DOS). Importantly, the data and analysis at high energy, $E$, show conservation of states: those removed from near $E_F$ are found between $\sim$30 and 70 meV from $E_F$. This quantum correction to the DOS agrees in magnitude with the $\sqrt{T}$ dependence of the bulk conductivity. Combining our results, with published theory and quantum interference data, we find a scattering time and Fermi velocity that agree reasonably well with recent ARPES results.