The determinant of one-dimensional polyharmonic operators of arbitrary order

摘要

We obtain an explicit expression for the regularised spectral determinant of the polyharmonic operator P-n = (-1)(n)(partial derivative(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 + [7 zeta(3)/2 pi(2)+ 3/2 + log )T/4)] n(2) + O(n) for large n. This is a consequence of sharp upper and lower bounds for log(det P-n) valid for all nand 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. (C) 2020 Elsevier Inc. All rights reserved.