Affiliations: LMPA Joseph Liouville ULCO, Université Lille Nord de France, Calais, France | Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona, Spain
Note: [] Corresponding author: Joachim von Below, LMPA Joseph Liouville ULCO, FR CNRS Math. 2956, Université Lille Nord de France, 50, rue F. Buisson, B.P. 699, F-62228 Calais Cedex, France. Tel.: +33 3 21 46 36 27; Fax: +33 3 21 46 36 69; E-mail: [email protected].
Abstract: We determine the asymptotic behavior of the eigenvalues of second-order elliptic operators on finite networks under continuity and weighted Kirchhoff flow conditions at the vertices. It turns out that the quadratic growth formula in the symmetrizable case [Linear Algebra Appl. 121 (1989), 692–697; Math. Meth. Appl. Sci. 10 (1988), 383–395; Parabolic Network Equations, 2nd edn, Universitätsverlag, Tübingen, 1994] holds also in the general case when the problem cannot be reduced to a self-adjoint problem and when non-real eigenvalues occur.
Keywords: elliptic operators on networks, eigenvalue asymptotics, adjacency and transition operators
