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European Journal of Physics

An exactly soluble Hartree problem in an external potential

J.C. Gunn, J M F Gunn

Published Jan 1, 1988• 10 citations
Publisher

Abstract

The problem of N bosons interacting with each other via repulsive delta-function interactions and with an external, attractive, delta-function potential is solved, within the Hartree approximation, exactly. If the interparticle interactions are above a certain value, there is no bound state. Thus the bound state does not just expand to compensate for the increase in the repulsive Hartree potential. Moreover as the interaction strength is increased to that value, the ground-state wavefunction develops a pole at the position of the attractive potential.

DOI
10.1088/0143-0807/9/1/009

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The problem of N bosons interacting with each other via repulsive delta-function interactions and with an external, attractive, delta-function potential is solved, within the Hartree approximation, exactly. If the interparticle interactions are above a certain value, there is no bound state. Thus the bound state does not just expand to compensate for the increase in the repulsive Hartree potential. Moreover as the interaction strength is increased to that value, the ground-state wavefunction develops a pole at the position of the attractive potential.
The exact solution for the ground state of the quantum one-dimensional NNN-boson problem with attractive $\ensuremath{\delta}$-function two-body potentials is compared with the (exact) self-consistent solution of the corresponding variational Hartree problem.