$ζ$-regularised spectral determinants on metric graphs

Details $ζ$-regularised spectral determinants on metric graphs $ζ$-regularised spectral determinants on metric graphs

2010-06-10

arxiv.org/abs/1006.2163v1

J. Phys. A: Math. Theor. 43 (2010) 425203

Physics

Mathematical Physics

LaTeX, 21 pages, 2 pdf figures

Scientific paper

10.1088/1751-8113/43/42/425203

Several general results for the spectral determinant of the Schr\"odinger operator on metric graphs are reviewed. Then, a simple derivation for the $\zeta$-regularised spectral determinant is proposed, based on the Roth trace formula. Two types of boundary conditions are studied: functions continuous at the vertices and functions whose derivative is continuous at the vertices. The $\zeta$-regularised spectral determinant of the Schr\"odinger operator acting on functions with the most general boundary conditions is conjectured in conclusion. The relation to the Ihara, Bass and Bartholdi formulae obtained for combinatorial graphs is also discussed.

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