An Inverse Spectral Result for the Neumann Operator on Planar Domains

The Neumann operator is an operator on the boundary of a smooth manifold which maps the boundary value of a harmonic function to its normal derivative. In this paper, the Neumann operator on the boundary of smooth, bounded, simply connected planar domains is studied. The asymptotics of the eigenvalues are computed. The regularised zeta function for the Neumann operator at z = − 2 is computed. Study of the zeta function is then used to show that the unit disk is characterised by the spectrum of its Neumann operator.