The Lame equation in parametric resonance after inflation

We show that the most general inflaton potential in Minkowski spacetime for which the differential equation for the Fourier modes of the matter fields reduces to Lam\'e's equation is of the form $V(\phi)= \lambda\phi^4/4+K\phi^2/2+\mu/(2\phi^2)+V_0$. As an application, we study the preheating phase after inflation for the above potential with K=0 and arbitrary $\lambda,\mu>0$. For certain values of the coupling constant between the inflaton and the matter fields, we calculate the instability intervals and the characteristic exponents in closed form.