  The beta-ensemble with cubic potential can be used to study a quantum
particle in a double-well potential with symmetry breaking term. The quantum
mechanical perturbative energy arises from the ensemble free energy in a novel
large N limit. A relation between the generating functions of the exact
non-perturbative energy, similar in spirit to the one of Dunne-Unsal, is found.
The exact quantization condition of Zinn-Justin and Jentschura is equivalent to
the Nekrasov-Shatashvili quantization condition on the level of the ensemble.
Refined topological string theory in the Nekrasov-Shatashvili limit arises as a
large N limit of quantum mechanics.
