The eigenvalues of the Laplacian on a sphere with boundary conditions specified on a segment of a great circle

We prove that the eigenvalues of the Laplacian on a sphere with a Dirichlet boundary condition specified on a segment of a great circle lie between an integer and a half-integer and for a Neumann boundary condition they lie between a half integer and an integer. These eigenvalues correspond to the eigenvalues of the angular part of the Laplacian with boundary conditions specified on a plane angular sector, which are relevant in the calculation of scattering amplitude. These eigenvalues can also be used to determine the behavior of the fields near the tip of a plane angular sector as a function of the distance to the tip. The first few eigenvalues for both Dirichlet and Neumann boundary conditions are calculated. The same eigenvalues are also calculated using the Wentzel–Kramers–Brillouin (WKB) method. There is excellent agreement between the exact and the WKB eigenvalues.