  We prove the smoothness of the L^2-analytic torsion form on some fiber
bundles with non-compact fibers of positive Novikov-Shubin invariant. We do so
by generalizing the arguments of Azzali-Goette-Schick to an appropriate Sobolev
space, and proving that the Novikov-Shubin invariant remains positive in the
Sobolev settings, using an argument of Alvarez Lopez-Kordyukov.
