Analytic singularities of the Bergman kernel for tubes
Let \(\Omega\subset\mathbb R^n\) be a bounded, convex, and open set with real analytic boundary. Let \(T_\Omega\subset\mathbb C^n\) be the tube with base \(\Omega\), and let \(\mathcal B(z,w)\) be the Bergman kernel of \(T_\Omega\). If \(\Omega\) is strongly convex, then \(\mathcal B(z,w)\) is analytic away from the boundary diagonal. In the weakly convex case this is no longer true. The authors show that off-diagonal singularities are precisely described by the characteristic lines. These lines are contained in the boundary of \(T_\Omega\), and they are projections of the Treves curves. These curves are symplectic invariants that are determined by the CR structure of the boundary of \(T_\Omega\). Note that Treves curves exist only when \(\Omega\) has at least one weakly convex boundary point. A brief discussion of the formula of L.~Boutet de Monvel for \(\mathcal B(z,w)\) is given. The precise version of the stationary phase formula that the authors need is proved. Using the method of Pólya, it is proved that the zeros of the entire function \(\mathcal N(\eta)=\int_{-\infty}^{+\infty}e^{2(\eta s-s^m)} ds\), \(m\geq 2\) is even, which are closest to the origin are simple ones.
analytic singularities away from the boundary diagonal
scientific article; zbMATH DE number 1820829
scientific article