The determinant of one-dimensional polyharmonic operators of arbitrary order

Abstract

We obtain an explicit expression for the regularised spectral determinant of the polyharmonic operator $P_{n}=(-1)^{n} (\partial_x)^{2n}$ on $(0,T)$ with Dirichlet boundary conditions and $n$ a positive integer, and show that it satisfies the asymptotics $\log{(\det P_{n})} = -n^2 \log{n} + \left[\frac{7\zeta(3)}{2\pi^2}+ \frac{3}{2}+\log\left(\frac{T}{4}\right)\right] n^2 + {\rm O}(n)$ for large $n$. This is a consequence of sharp upper and lower bounds for $\log{(\det P_{n})}$ valid for all $n$ and which coincide in the terms up to order $n$. These results form the basis to analyse more general operators with nonconstant coefficients and show that the corresponding determinants have a similar asymptotic behaviour.