  Generalized torical band inequalities give precise upper bounds for the width
of compact manifolds with boundary in terms of positive pointwise lower bounds
for scalar curvature, assuming certain topological conditions. We extend
several incarnations of these results in which pointwise scalar curvature
bounds are replaced with spectral scalar curvature bounds. More precisely, we
prove upper bounds for the width in terms of the principal eigenvalue of the
operator $-\Delta +cR$, where $R$ denotes scalar curvature and $c>0$ is a
constant. Three separate strategies are employed to obtain distinct results
holding in different dimensions and under varying hypotheses, namely we utilize
spacetime harmonic functions, $\mu$-bubbles, and spinorial Callias operators.
In dimension 3, where the strongest result is produced, we are also able to
treat open and incomplete manifolds, and establish the appropriate rigidity
statements. Additionally, a version of such spectral torus band inequalities is
given where tori are replaced with cubes. Finally, as a corollary we generalize
classical work of Schoen and Yau, on the existence of black holes due to
concentration of matter, to higher dimensions and with alternate measurements
of size.
