Abstract
Approximate but reliable solutions of a quantum system with $N$ identical particles can be easily computed with the envelope theory, also known as the auxiliary field method. This technique has been developed for Hamiltonians with arbitrary kinematics and one- or two-body potentials. It is adapted here for cyclic systems with $N$ identical particles, that is to say systems in which a particle $i$ has only an interaction with particles $i-1$ and $i+1$ (with $N+1\equiv 1$).
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