Eigenvalue asymptotics for Schrödinger operators on sparse graphs

Michel Bonnefont (1) , Sylvain Golenia (1) , Matthias Keller (2)

1 IMB - Institut de Mathématiques de Bordeaux
2 Mathematisches Institut, Friedrich Schiller Universität Jena

We consider Schrödinger operators on sparse graphs. The geometric definition of sparseness turn out to be equivalent to a functional inequality for the Laplacian. In consequence, sparseness has in turn strong spectral and functional analytic consequences. Specifically, one consequence is that it allows to completely describe the form domain. Moreover, as another consequence it leads to a characterization for discreteness of the spectrum. In this case we determine the first order of the corresponding eigenvalue asymptotics.

Mots clés: discrete laplacian; eigenvalues; asymptotic; planarity; sparse; functional inequality

Domaines: Théorie spectrale [math.SP]; Analyse fonctionnelle [math.FA]

Michel Bonnefont, Sylvain Golenia, Matthias Keller. Eigenvalue asymptotics for Schrödinger operators on sparse graphs. 2014. ⟨hal-00910384v2⟩