We study a physically motivated model for a trapped dilute gas of Bosons with repulsive pairwise atomic interactions at zero temperature. Our goal is to describe aspects of the excited many-body quantum states of this system by accounting for the scattering of atoms in pairs from the macroscopic state. We start with an approximate many-body Hamiltonian,
H
a
p
p
\mathcal {H}_{\mathrm {app}}
, in the Bosonic Fock space. This
H
a
p
p
\mathcal {H}_{\mathrm {app}}
conserves the total number of atoms. Inspired by Wu [J. Math. Phys. 2 (1961), 105–123], we apply a non-unitary transformation to
H
a
p
p
\mathcal {H}_{\mathrm {app}}
. Key in this procedure is the pair-excitation kernel, which obeys a nonlinear integro-partial differential equation. In the stationary case, we develop an existence theory for solutions to this equation by a variational principle. We connect this theory to a system of partial differential equations for one-particle excitation (“quasiparticle”-) wave functions derived by Fetter [Ann. Phys. 70 (1972), 67–101], and prove existence of solutions for this system. These wave functions solve an eigenvalue problem for a
J
J
-self-adjoint operator. From the non-Hermitian Hamiltonian, we derive a one-particle nonlocal equation for low-lying excitations, describe its solutions, and recover Fetter’s energy spectrum. We also analytically provide an explicit construction of the excited eigenstates of the reduced Hamiltonian in the
N
N
-particle sector of Fock space.