In the this article by Lieb, Liniger: "Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State" a repulsive Bose gas is considered in 1d with Hamiltonian $$ H = -\sum_\ell \partial_\ell^2 + 2c \sum_\ell \sum_{m<\ell} \delta(x_m-x_\ell) $$

In one passage the solution space is considered for positive and negative $c$. It is argued:

we are interested in the repulsive case $c\geq0$. While the attractive case $c<0$ has solutions, the case is not physically meaningful, because there is no saturation. It can be shown that the energy of the $N$ particle ground state is proportional to $-N$ for $c\geq0$ but $-N^2$ for $c<0$.

I am interested in understanding the physical meaning of this particle-energy proportionality, and what the meaning is of this saturation?


1 Answer 1


I suspect that what was meant by that statement (lacking saturation) is that no stable N-body state exists for the attractive case. For fermion systems the attractive case can be stabilized by the Pauli principle, but Bose systems lack this stabilization mechanism. Leib"s work has for many decades specialized in the study of stability in many body quantum systems.


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