Heat asymptotics on quantum graphs

• Applicant: Dr. Ralf Rueckriemen
• Subject Area: Mathematics
• Term: from 2012 to 2013
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 209743911

[see attached files for a version with mathematical formulas] It is well known that the heat kernel of the Laplacian on a manifold admits an asymptotic expansion . This has been generalized to operators of the form Delta +V on manifolds with boundary. It takes the form\begin{equation*} \sum_{\lambda_i}e^{-\lambda_i t} \sim_{t \ra 0^+} (4\pi t)^{-\frac{n}{2}}\sum_{k=0}^{\infty}a_kt^{\frac{k}{2}} + (4\pi t)^{-\frac{n-1}{2}}\sum_{k =0}^{\infty}b_kt^{\frac{k}{2}}\end{equation*}for some coefficients $a_k=\int_M \tilde{a}_k(x)dx$ and $b_k=\int_{\partial M} \tilde{b}_k(x)dx$ that are universal in the curvature and the potential. It is possible to compute the first few of these coefficients explicitly.We plan to prove a similar formula for quantum graphs. As quantum graphs do not have any curvature the case of the standard Laplacian would just give an asymptotic approximation of the trace formula. However in the more general case of operators of the form Delta+V with a potential function there is no hope for a trace formula because the eigenvalue problem cannot be solved explicitly on the individual edges. Thus heat asymptotics are the best option, they will still carry some spectrally determined information about the potential.The existence of the asymptotics on manifolds without boundary can be shown through a local parametrix. The more general case of manifolds with boundary is shown through an inversion of the heat operator on the level of symbols. We are hopeful that this method can be adapted to the setting of quantum graphs. The explicit computation of the first few coefficients is done through some recursion relations. It is based on the observation that the operators (d + alpha)^*(d + alpha) and (d + alpha)(d + alpha)^* have the same spectrum. If one considers operators acting on functions and on 1-forms this approach is adaptable to quantum graphs.

• DFG Programme: Research Fellowships
• International Connection: United Kingdom
• Host: Professor Dr. Jens Bolte