Time-dependent harmonic oscillator and spectral determinant on graphs

Abstract:

Using a path integral approach and also considerations about the time-dependent harmonic oscillator, we compute the spectral determinant of the operator () on a graph. ( is the Laplacian and V(x) is some potential defined on the graph). We recover a recent result that was obtained by constructing the Green's function on the graph. We also extend those considerations to the case when i) a magnetic field is added to the system, ii) the potential, V(x), contains repulsive peaks.