Heat-kernel and resolvent asymptotics for Schroedinger operators on   metric graphs
  We consider Schroedinger operators on compact and non-compact (finite) metric graphs. For such operators we analyse their spectra, prove that their resolvents can be represented as integral operators and introduce trace-class regularisations of the resolvents. Our main result is a complete asymptotic expansion of the trace of the (regularised) heat-semigroup generated by the Schroedinger operator. We also determine the leading coefficients in the expansion explicitly. 
Jens Bolte, Sebastian Egger, Ralf Rueckriemen
math-ph math.MP math.SP