Mathematical Physics and Harmonic Analysis Seminar
31 August 2007, Bloc 627

Title: Index theorems for quantum graphs
Speaker: Stephen A. Fulling, TAMU

Abstract:
In geometric analysis, an index theorem relates the difference of 
the numbers of solutions of two differential equations to the 
topological structure of the manifold or bundle concerned, 
sometimes using the heat kernels of two higher-order differential 
operators as an intermediary. In this work, joint with Peter 
Kuchment and Justin Wilson, the case of quantum graphs is 
addressed. A quantum graph is a graph considered as a (singular) 
one-dimensional variety and equipped with a second-order 
differential Hamiltonian $H$ (a~``Laplacian'') with suitable 
conditions at vertices. For the case of scale-invariant vertex 
conditions (i.e., conditions that do not mix the values of 
functions and of their derivatives), the constant term of the 
heat-kernel expansion is shown to be proportional to the trace of 
the internal scattering matrix of the graph. This observation is 
placed into the index-theory context by factoring the Laplacian 
into two first-order operators, $H=A^*A$, and relating the 
constant term to the index of~$A$. An independent consideration 
provides an index formula for any differential operator on a 
finite quantum graph in terms of the vertex conditions. It is 
found also that the algebraic multiplicity of $0$ as a root of 
the secular determinant of $H$ is the sum of the nullities of $A$ 
and $A^*$.