Bifurcation of double eigenvalues for Aharonov–Bohm operators with a moving pole

Abstract

We study double eigenvalues of Aharonov–Bohm operators with Dirichlet boundary conditions in planar domains containing the origin. We focus on the behavior of double eigenvalues when the potential’s circulation is a fixed half-integer number and the operator’s pole is moving on straight lines in a neighborhood of the origin. We prove that bifurcation occurs if the pole is moving along straight lines in a certain number of cones with positive measure. More precise information is given for symmetric domains; in particular, in the special case of the disk, any eigenvalue is double if the pole is located at the center, but there exists a whole neighborhood where it bifurcates into two distinct branches.