  We investigate the effect of a non-uniform deformation applied to
one-dimensional (1D) quantum systems, where the local energy scale is
proportional to $g_j = [\sin (j \pi / N)]^m$ determined by a positive integer
$m$, site index $1 \leq j \leq N-1$, and the system size N. This deformation
introduces a smooth boundary to systems with open boundary conditions. When $m
\geq 2$, the leading $1/N$ correction to the ground state energy per bond
$e_0^{(N)}$ vanishes and one is left with a $1/N^2$ correction, the same as
with periodic boundary conditions. In particular, when $m = 2$, the value of
$e_0^{(N)}$ obtained from the deformed open-boundary system coincides with the
uniform system with periodic boundary conditions. We confirm the fact
numerically for correlated systems, such as the extended Hubbard model, in
addition to 1D free-Fermion models.
