Heat asymptotics on quantum graphs
Final Report Abstract
I proved existence of the heat kernel in a fairly general setting using known properties of the eigenfunctions of the operator on an interval. Using the parametrix method, I gave a general form of the asymptotic expansion of the heat kernel and computed the first couple of coefficients explicitly. Using the parametrix method requires some assumptions on the behaviour of the potential at the vertices. In joint work with Uzy Smilansky we proved a trace formula in this setting. It relates the eigenvalues to topological properties of the graph and the exact solutions of the eigenvalue equation on the individual edges. As these solutions are not explicitly known we also proved an approximate version that replaces the exact solutions by a WKB-approximation. Finally, I am currently working with Sebastian Eggers on using the resolvent to study the heat kernel. One can write down an explicit formula for the resolvent in a very general setting and show it admits an asymptotic expansion. We then used a Laplace transform to get an asymptotic expansion of the heat kernel. This method so far seems to work in the most general setting but explicitly computing coefficients beyond the first two or three seems hard.
Publications
- Trace formulae for quantum graphs with edge potentials, 2012, Journal of Physics A, Mathematical and Theoretical, 45 (2012) 475205
Ralf Rueckriemen and Uzy Smilansky
- Heat trace asymptotics for quantum graphs
Ralf Rueckriemen
