QFT over the finite line. Heat kernel coefficients, spectral zeta functions and selfadjoint extensions

Following the seminal works of Asorey-Ibort-Marmo and Mu\~{n}oz-Casta\~{n}eda-Asorey about selfadjoint extensions and quantum fields in bounded domains, we compute all the heat kernel coefficients for any strongly consistent selfadjoint extension of the Laplace operator over the finite line $[0,L]$. The derivative of the corresponding spectral zeta function at $s=0$ (partition function of the corresponding quantum field theory) is obtained. In order to compute the correct expression for the $a_{1/2}$ heat kernel coefficient, it is necessary to know in detail which non-negative selfadjoint extensions have zero modes and how many of them they have. The answer to this question leads us to analyse zeta function properties for the Von Neumann-Krein extension, the only extension with two zero modes.