Isolated solutions of a local polaron model

A simple local polaron model is studied. The Schrödinger equation for the model in the holomorphic representation is reduced to the system of the first order differential equations. The eigen-values are determined from the condition that the solution belongs to the class of holomorphic functions. In the general case, the eigen-values are found from a certain transcendental equation including continuous fraction. On the basis of the general theory of differential equations algebraic equations are obtained for the eigen-vectors of the simplest isolated solutions. Possibility of intersection of eigenvalues in isolated points is demonstrated. Impossibility of phase transition for the simple local polaron model is rigorously proved.