Schrodinger operators with magnetic fields and minimal action functionals

We consider a convex superlinear Lagrangian L on a closed connected manifol d M such that its associated Hamiltonian H satisfies controllable growth co nditions. For this class of Lagrangians we define H-harmonic functions and the harmonic value h and we compare it with Mane's critical value c. We sho w, using elliptic regularity of quasilinear elliptic equations, that h less than or equal to c and the equality holds iff there exists a unique (up to constants) smooth weak KAM solution which is H-harmonic. Fix a Riemannian metric on M with volume one and consider a real C-infinity 1-form theta and a smooth function V: M --> R. Let L be the convex and sup erlinear Lagrangian given by \ L(x, nu) := 1/2/nu/(2)(x) + theta (x)(nu) - V(x). This is a special but important class of Lagrangians. We consider the Schro dinger operator H((theta ,V)) associated with L and we let lambda (0) be it s first eigenvalue. We show that lambda (0) less than or equal to h with eq uality only if h = c. When h = c this common value is an eigenvalue of H-(t heta ,H-V), but not necessarily the smallest one. Using these ideas we defi ne a norm / (.) /(Schr) in H-1 (M,R) that we call the Schrodinger norm and we compare it with the L-2 norm / (.) /(L2) and with the stable norm / (.) /(s). We show that for any cohomology class [omega] epsilon H-1 (M,R) we ha ve /[omega]/(Schr) less than or equal to /[omega]/(L2) less than or equal to / [omega]/(s). Any of the inequalities is an equality if and only if the unique harmonic r epresentative in [omega] has constant Riemannian norm. We derive various co rollaries from these results.