  A subtle relation between Quantum Hall physics and the phenomenon of pairing
is unveiled. By use of second quantization, we establish a connection between
(i) a broad class of rotationally symmetric two-body interactions within the
lowest Landau level and (ii) integrable hyperbolic Richardson-Gaudin type
Hamiltonians that arise in (p_{x}+ip_{y}) superconductivity. Specifically, we
show that general Haldane pseudopotentials (and their sums) can be expressed as
a sum of repulsive non-commuting (p_{x}+ip_{y})-type pairing Hamiltonians. For
the Laughlin sequence, it is observed that this problem is frustration free and
zero energy ground states lie in the common null space of all of these
non-commuting Hamiltonians. This property allows for the use of a new truncated
basis of pairing configurations in which to express Laughlin states at general
filling factors. We prove separability of arbitrary Haldane pseudopotentials,
providing explicit expressions for their second quantized forms, and further
show by explicit construction how to exploit the topological equivalence
between different geometries (disk, cylinder, and sphere) sharing the same
topological genus number, in the second quantized formalism, through similarity
transformations. As an application of the second quantized approach, we
establish a "squeezing principle" that applies to the zero modes of a general
class of Hamiltonians, which includes but is not limited to Haldane
pseudopotentials. We also show how one may establish (bounds on)
"incompressible filling factors" for those Hamiltonians. By invoking properties
of symmetric polynomials, we provide explicit second quantized quasi-hole
generators; the generators that we find directly relate to bosonic chiral edge
modes and further make aspects of dimensional reduction in the Quantum Hall
systems precise.
