Instability of sphalerons in $\phi^4$ models with a false vacuum

A one-parameter family of nonlinear (quartic) Klein-Gordon models having a sphaleron solution is studied. The sphaleron arises from a saddle point between true and false vacua in the energy functional. Its instability is shown be governed by a Heun differential equation after a change of variable. This allows an explicit formulation of the eigenfunctions and eigenvalues to be obtained in terms of local Heun functions. Good approximations are found for certain ranges of the parameter.