# Functional determinants

Spectral determinants (functional determinants) on metric graphs

The spectral determinant is a compact object encoding information on the spectrum of a linear operator (the Laplace or the Schrödinger operator, of interest here):

S(γ)=det(γ-Δ+V(x))=∏(γ+En)

where γ is a (spectral) parameter and {En} is the spectrum of eigenvalues of the operator -Δ+V(x) (the product runs over all eigenstates ; it is a formal writing since the set of eigenstates is infinite). Such objects are also denoted « functional determinants » in the mathematical literature.

A trace formula for the partition function (heat kernel) of the Laplace operator on a metric graph was obtained by J.-P. Roth in 1983. In our article we have established the relation between the Roth’s trace formula and a recent result by M. Pascaud & G. Montambaux expressing the spectral determinant in term of the determinant of a matrix of finite size coupling vertices of the graph.

Spectral determinants of metric graphs glued at one point

The spectral determinant is a compact object that can be easily and systematically constructed for metric graphs in terms of their topological properties. This short paper shows that when two graphs G1 and G2 are glued at one point in order to form a larger graph G, Spectral determinant of G can be related to spectral determinants of G1 and G2. Therefore spectrum of G can be related to the spectra of G1 and G2 (plus some information on the connectivity of the vertex where graphs are attached).

Zeta regularisation of spectral determinants on metric graphs

The question of the regularisation of the spectral determinant and its precise prefactor is discussed.

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