Determinants of Laplacians in Exterior Domains
By Andrew Hassell, Steve Zelditch. 2000
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We consider classes of simply connected planar domains which are isophasal, ie, have the same scattering phase $s(\l)$ for all $\l > 0$. This is a scattering-theoretic analogue of isospectral domains. Using the heat invariants and the determinant of the Laplacian, Osgood, Phillips and Sarnak showed that each isospectral class is sequentially compact in a natural $C$-infinity topology. In this paper, we show sequential compactness of each isophasal class of domains. To do this we define the determinant of the exterior Laplacian and use it together with the heat invariants (the heat invariants and the determinant being isophasal invariants). We show that the determinant of the interior and exterior Laplacians satisfy a Burghelea-Friedlander-Kappeler type surgery formula. This allows a reduction to a problem on bounded domains for which the methods of Osgood, Phillips and Sarnak can be adapted.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/dda69131cb0ebe5e08829d189a6d7f5f9c466c69
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In Mathematical Physics
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- Tau-functions on spaces of Abelian differentials and higher genus generalizations of Ray-Singer formula (2004, background): “Vol(L,gc)−logl(∂D) . (4.33) Proof. Formula (4.33) can be proved in the same manner as the formula (4.31) for smooth metrics. We shall follow the strategy of the original work [3] and the recent paper [14] devoted to determinants of Laplacians in exterior domains. 1 As in [3] and [14] the proof consi”