In the model of hole
superconductivity the strength of the pairing interaction depends on the
local carrier density. This gives rise to a dependence of the gap
function $\Delta_k$ on the band energy $\epsilon_k$. Fluctuations in the
local potential energy will result in different values of $\Delta_k$ at
the Fermi energy and hence in different values of the local energy gap.
In particular, the energy gap can be sharply reduced. We study this
behavior by numerical solution of the Bogoliubov-de Gennes equations for
the model. The behavior is contrasted with what occurs in the attractive
Hubbard model, where local potential fluctuations have negligible effect.
The physical origin of this behavior and the possible relevance to high
$T_c$ oxides is discussed.
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