Eigenfunctions of the essential spectrum of the Laplace operator in an angle with the Robin–Neumann boundary conditions

This work studies eigenfunction problem of the Laplace operator in the angular domain with the Robin-type boundary condition on the upper side of the angle and the Neumann-type boundary condition on the bottom side of the angle. From the physical point of view, such eigenfunctions describe waves over sloping beach. Negative values of the spectral parameter were considered. We obtained the eigenfunction of the essential spectrum and studied a special case of eigenfunction, which are elementary functions. The Sommerfeld integral representation of an eigenfunction of the negative part of the essential spectrum of the Laplace operator was obtained. Moreover, we calculated it's asymptotic far away from the angle's vertex. It is bounded on the top side of the angle and vanishes exponentially in the angle's interior with its bottom side. So, the eigenfunction of essential spectrum behaves like a surface wave.