  We study the low-energy spectral properties of positive center-of-mass
conserving two-body Hamiltonians as they arise in models of fractional quantum
Hall states. Starting from the observation that positive many-body Hamiltonians
must have ground-state energies that increase monotonously in particle number,
we explore what general additional constraints can be obtained for two-body
interactions with "center-of-mass conservation" symmetry, both in the presence
and absence of particle-hole symmetry. We find general bounds that constrain
the evolution of the ground-state energy with particle number, and in
particular, constrain the chemical potential at $T=0$. Special attention is
given to Hamiltonians with zero modes, in which case similar bounds on the
first excited state are also obtained, using a duality property. In this case,
in particular, an upper bound on the charge gap is also obtained. We further
comment on center of mass and relative decomposition in disk geometry within
the framework of second quantization.
