1-D Schrödinger operator on a star graph with nondefinite weight function

Abstract

On a star graph $G$ with $n=n_{+}+n_{-}$ edges of unit length, we study the operator $-\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}$ on $n_{+}$ and $\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}$ on $n_{-}$ edges equipped with Dirichlet boundary conditions at the outer vertices and a Kirchhoff condition at the central vertex. We study the spectral properties of the corresponding indefinite Kirchhoff Laplacian on $G$ and we show that it is similar to a selfadjoint operator in the Hilbert space $L^2(G)$ and that its eigenfunctions form a Riesz basis. Furthermore, we give a complete description of the point spectrum.