Discretization of zeta-determinants of Schrödinger operators on the torus. (Discrétisation de zeta-déterminants d’opérateurs de Schrödinger sur le tore.) (French) Zbl 1112.58035
Summary: We propose two results concerning the \(\zeta\)-regularised determinant \(\det_\zeta\) \(A\) of a Schrödinger operator \(A=\Delta_g+ V\) on a compact Riemannian manifold \(({\mathcal M},g)\). For \({\mathcal M}= S^1\times S^1\), we construct a sequence \((G_n,\rho_n,\Delta_n)\) where \(G_n\) is a finite graph injected in \({\mathcal M}\) via \(\rho_n\), in such a way that \(\rho_n(G_n)\) triangulates \({\mathcal M}\). \(\Delta_n\) is a discrete Laplacian on \(G_n\) so that for every potential \(V\) on \({\mathcal M}\), the sequence \(\det(\Delta_n+V)\) converges, after normalisation, to \(\det_\zeta(\Delta_g+V)\). Last, we give on every Riemannian compact manifold \(({\mathcal M},g)\) whose dimension is less than or equal to 3 and with a transitiv isometry group, the maximum of the determinant \(\det_\zeta(\Delta_g+V)\).
MSC:
| 58J52 | Determinants and determinant bundles, analytic torsion |
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 53C24 | Rigidity results |
| 94C15 | Applications of graph theory to circuits and networks |
| 58J40 | Pseudodifferential and Fourier integral operators on manifolds |
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
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