Zeta functions with Dirichlet and Neumann boundary conditions for exterior domains

Jean-Pierre Eckmann (1, 2) , Claude-Alain Pillet (3, 4, 5)

1 Département de Physique Théorique
2 Section de mathématiques [Genève]
3 CPT - E5 Physique statistique et systèmes complexes
4 CPT - Centre de Physique Théorique - UMR 6207
5 CPT - Centre de Physique Théorique - UMR 7332

We generalize earlier studies on the Laplacian for a bounded open domain $\Omega\subset\mathbb R^2$ with connected complement and piecewise smooth boundary. We compare it with the quantum mechanical scattering operator for the exterior of this same domain. Using single layer and double layer potentials we can prove a number of new relations which hold when one chooses independently Dirichlet or Neumann boundary conditions for the interior and exterior problem. This relation is provided by a very simple set of $\zeta$-functions, which involve the single and double layer potentials. We also provide Krein spectral formulas for all the cases considered and give a numerical algorithm to compute the $\zeta$-function.

Mots clés: Krein formula; zeta function; quantum billiards; inside-outside duality; scattering theory

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]; Théorie spectrale [math.SP]

Jean-Pierre Eckmann, Claude-Alain Pillet. Zeta functions with Dirichlet and Neumann boundary conditions for exterior domains. Helvetica Physica Acta, 1997, 70, pp.44-65. ⟨hal-00129124⟩