Spectral determinants and zeta functions of Schrödinger operators on metric graphs

Abstract

A derivation of the spectral determinant of the Schrödinger operator on a metric graph is presented where the local matching conditions at the vertices are of the general form classified according to the scheme of Kostrykin and Schrader. To formulate the spectral determinant we first derive the spectral zeta function of the Schrödinger operator using an appropriate secular equation. The result obtained for the spectral determinant is along the lines of the recent conjecture.

16 pages, 2 figures

References in corpus (5)

• Functionals of the Brownian motion, localization and metric graphs
• Computation of determinants using contour integrals
• Pistons modeled by potentials
• Quantum oscillations in mesoscopic rings and anomalous diffusion
• On the spectrum of the Laplace operator of metric graphs attached at a vertex -- Spectral determinant approach

Cited by in corpus (5)

• One-dimensional disordered quantum mechanics and Sinai diffusion with random absorbers
• Non-linear conductance in mesoscopic weakly disordered wires -- Interaction and magnetic field asymmetry
• Zeta Functions of the Dirac Operator on Quantum Graphs
• Relating Zeta Functions of Discrete and Quantum Graphs
• Can One Hear the Spanning Trees of a Quantum Graph?