  We consider the quantum spectral problem appearing the Fermi gas formulation
of the ABJM (Aharony-Bergman-Jafferis-Maldacena) matrix model. This is known to
related to the refined topological string on local P^1*P^1 Calabi-Yau geometry.
In the ABJM setting the problem is formulated by an integral equation, and is
somewhat different from the one formulated directly in terms of the Calabi-Yau
geometry and studied in our earlier paper. We use the similar method in our
earlier paper to determine the non-perturbative contributions to the quantum
phase volume in the ABJM case from the Bohr-Sommerfeld quantization condition.
As in our earlier paper, the non-perturbative contributions contain higher
order smooth corrections beyond those required by singularity cancellations
with the perturbative contributions proposed by Kallen and Marino. Our results
imply possible new contributions to the grand potential of the ABJM matrix
model.
