  We study the one-dimensional nearest neighbor tight binding model of
electrons with independently distributed random hopping and no on-site
potential (i.e. off-diagonal disorder with particle-hole symmetry, leading to
sub-lattice symmetry, for each realization). For non-singular distributions of
the hopping, it is known that the model exhibits a universal, singular behavior
of the density of states $\rho(E) \sim 1/|E \ln^3|E||$ and of the localization
length $\xi(E) \sim |\ln|E||$, near the band center $E = 0$. (This singular
behavior is also applicable to random XY and Heisenberg spin chains; it was
first obtained by Dyson for a specific random harmonic oscillator chain).
Simultaneously, the state at $E = 0$ shows a universal, sub-exponential decay
at large distances $\sim \exp [ -\sqrt{r/r_0} ]$. In this study, we consider
singular, but normalizable, distributions of hopping, whose behavior at small
$t$ is of the form $\sim 1/ [t \ln^{\lambda+1}(1/t) ]$, characterized by a
single, continuously tunable parameter $\lambda > 0$. We find, using a
combination of analytic and numerical methods, that while the universal result
applies for $\lambda > 2$, it no longer holds in the interval $0 < \lambda <
2$. In particular, we find that the form of the density of states singularity
is enhanced (relative to the Dyson result) in a continuous manner depending on
the non-universal parameter $\lambda$; simultaneously, the localization length
shows a less divergent form at low energies, and ceases to diverge below
$\lambda = 1$. For $\lambda < 2$, the fall-off of the $E = 0$ state at large
distances also deviates from the universal result, and is of the form $\sim
\exp [-(r/r_0)^{1/\lambda}]$, which decays faster than an exponential for
$\lambda < 1$.
