Eigenvalue asymptotics for second-order elliptic operators on networks

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.