# On the spectrum of the multiplicative Hilbert matrix

## TL;DR

This paper investigates the spectral properties of the multiplicative Hilbert matrix, establishing the nature of its spectrum and spectral multiplicity using advanced spectral and scattering theory techniques.

## Contribution

It proves the absence of singular continuous spectrum and confirms the simple multiplicity of the absolutely continuous spectrum for the multiplicative Hilbert matrix.

## Key findings

- No singular continuous spectrum for the matrix
- Absolutely continuous spectrum has multiplicity one
- Spectral properties analyzed using perturbation and scattering theory

## Abstract

We study the multiplicative Hilbert matrix, i.e. the infinite matrix with entries $(\sqrt{mn}\log(mn))^{-1}$ for $m,n\geq2$. This matrix was recently introduced within the context of the theory of Dirichlet series, and it was shown that the multiplicative Hilbert matrix has no eigenvalues and that its continuous spectrum coincides with $[0,\pi]$. Here we prove that the multiplicative Hilbert matrix has no singular continuous spectrum and that its absolutely continuous spectrum has multiplicity one. Our argument relies on the tools of spectral perturbation theory and scattering theory. Finding an explicit diagonalisation of the multiplicative Hilbert matrix remains an interesting open problem.