Spectral curves and the Schroedinger equations for the Eynard-Orantin recursion

It is predicted that the principal specialization of the partition function of a B-model topological string theory, that is mirror dual to an A-model enumerative geometry problem, satisfies a Schroedinger equation, and that the characteristic variety of the Schroedinger operator gives the spectral curve of the B-model theory, when an algebraic K-theory obstruction vanishes. In this paper we present two concrete mathematical A-model examples whose mirror dual partners exhibit these predicted features on the B-model side. The A-model examples we discuss are the generalized Catalan numbers of an arbitrary genus and the single Hurwitz numbers. In each case, we show that the Laplace transform of the counting functions satisfies the Eynard-Orantin topological recursion, that the B-model partition function satisfies the KP equations, and that the principal specialization of the partition function satisfies a Schroedinger equation whose total symbol is exactly the Lagrangian immersion of the spectral curve of the Eynard-Orantin theory.