Critical Casimir forces for O(n) systems with long-range interaction in the spherical limit.
We present exact results on the behavior of the thermodynamic Casimir force and the excess free energy in the framework of the d -dimensional spherical model with a power law long-ranged interaction decaying at large distances r as r(-d-sigma) , where sigma T(c) and T< T(c) . The universal finite-size scaling function governing the behavior of the force in the critical region is derived and its asymptotics are investigated. While in the critical and subcritical region the force is of the order of L(-d) , for T> T(c) it decays as L(-d-sigma) , where L is the thickness of the film. We consider both the case of a finite system that has no phase transition of its own, when d-1 sigma , when one observes a dimensional crossover from d to a d-1 dimensional critical behavior. The behavior of the force along the phase coexistence line for a magnetic field H=0 and T< T(c) is also derived. We have proven analytically that the excess free energy is always negative and monotonically increasing function of T and H . For the Casimir force we have demonstrated that for any sigma > or =1 it is everywhere negative, i.e., an attraction between the surfaces bounding the system is to be observed. At T= T(c) the force is an increasing function of T for sigma>1 and a decreasing one for sigma<1 . For any d and sigma the minimum of the force at T= T(c) is always achieved at some H unequal to 0 .