Another look at qualitative properties of eigenvalues using effective Hamiltonians

The goal of this paper is to review several qualitative properties of well-known eigenvalue problems using a different perspective based on the theory of effective Hamiltonians, working exclusively on the Hopf-Cole transform of the equation. We revisit some monotonicity results as well as the derivation of several scaling limits by means of the Donsker-Varadhan formula, and we point out several differences between the case of quadratic Hamiltonians and non-quadratic ones.