Many-body excitations in trapped Bose gas: A non-Hermitian approach

Grillakis, Manoussos; Margetis, Dionisios; Sorokanich, Stephen

doi:10.1090/qam/1630

Authors: Manoussos Grillakis, Dionisios Margetis and Stephen Sorokanich
Journal: Quart. Appl. Math. 81 (2023), 87-126
MSC (2020): Primary 35Q40, 81V73, 81Q12; Secondary 47N20, 82C10
DOI: https://doi.org/10.1090/qam/1630
Published electronically: September 26, 2022
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Abstract: 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, $\mathcal {H}_{\mathrm {app}}$, in the Bosonic Fock space. This $\mathcal {H}_{\mathrm {app}}$ conserves the total number of atoms. Inspired by Wu [J. Math. Phys. 2 (1961), 105–123], we apply a non-unitary transformation to $\mathcal {H}_{\mathrm {app}}$. 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 $J$-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 $N$-particle sector of Fock space.

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