The -Neumann operator on (0, q)-forms (1 q n) on a bounded convex domain 0 in C n is compact if and only if the boundary of 0 contains no complex analytic (equivalently: affine) variety of dimension greater than or equal to q. 1998Academic Press

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We study the asymptotic behavior, in a "semi-classical limit", of the first eigenvalues (i.e. the groundstate energies) of a class of Schrödinger operators with magnetic fields and the relationship of this behavior with compactness in thē ∂-Neumann problem on Hartogs domains in C 2 .

Abstract. We show how to compute the Bergman kernel functions of some special domains in a simple way. As an application of the explicit formulas, we show that the Bergman kernel functions of some convex domains, for instance the domain in C 3 defined by the inequality |z 1 | + |z 2 | + |z 3 | < 1, have zeroes.