Zero modes of quantum graph Laplacians and an index theorem

Abstract

We study zero modes of Laplacians on compact and non-compact metric graphs with general self-adjoint vertex conditions. In the first part of the paper the number of zero modes is expressed in terms of the trace of a unitary matrix $\mathfrak{S}$ that encodes the vertex conditions imposed on functions in the domain of the Laplacian. In the second part a Dirac operator is defined whose square is related to the Laplacian. In order to accommodate Laplacians with negative eigenvalues it is necessary to define the Dirac operator on a suitable Kre\uın space. We demonstrate that an arbitrary, self-adjoint quantum graph Laplacian admits a factorisation into momentum-like operators in a Kre\uın-space setting. As a consequence, we establish an index theorem for the associated Dirac operator and prove that the zero-mode contribution in the trace formula for the Laplacian can be expressed in terms of the index of the Dirac operator.

References in corpus (5)

• Spectra of self-adjoint extensions and applications to solvable Schroedinger operators
• The trace formula for quantum graphs with general self adjoint boundary conditions
• Index theorems for quantum graphs
• The inverse scattering problem for metric graphs and the traveling salesman problem
• Finite propagation speed and causal free quantum fields on networks

Cited by in corpus (2)

• Quantizing graphs, one way or two?
• Many-particle quantum graphs: A review