The spectral shift function and Levinson's theorem for quantum star   graphs
  We consider the Schr\"odinger operator on a star shaped graph with $n$ edges joined at a single vertex. We derive an expression for the trace of the difference of the perturbed and unperturbed resolvent in terms of a Wronskian. This leads to representations for the perturbation determinant and the spectral shift function, and to an analog of Levinson's formula. 
Semra Demirel
math.SP math-ph math.MP