Abstract:
One-dimensional spherical model with slowly decreasing and oscillating potential of the form $\rho(r)=r^{-1}\sin\alpha r$ is considered. The dependence of the free energy upon the boundary conditions is studied. It turns out that in the case of zero (finite, in general) boundary conditions the free energy $\psi_0(\beta)$ is analytical for all $\beta>0$. In the case of periodical boundary conditions the free energy $\psi(\beta)$ coincides with $\psi_0(\beta)$ for small $\beta$'s. However at some points $\beta_c$ the new branches of the free energy arise. Therefore in this situation the standard method of analytical continuation from the domain of small $\beta$'s is not applicable so far as it does not catch the phase transition.
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