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



Author: Christophe Texier

Source: https://arxiv.org/







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