Semi-classical analysis of Schrodinger operators and compactness in the d-bar-Neumann problem

We study the asymptotic behavior, in a ``semi-classical limit’', of the first eigenvalues (i.e. the groundstate energies) of a class of Schr"{o}dinger operators with magnetic fields and the relationship of this behavior with compactness in the $\bar{\mathrm{\partial }}$-Neumann problem on Hartogs domains in ${\text{C}}^2$