We consider a magnetic operator of Aharonov–Bohm type with Dirichlet boundary
conditions in a planar domain. We analyze the behavior of its eigenvalues as the
singular pole moves in the domain. For any value of the circulation we prove that the
-th magnetic eigenvalue
converges to the
-th
eigenvalue of the Laplacian as the pole approaches the boundary. We show that the
magnetic eigenvalues depend in a smooth way on the position of the pole, as long as
they remain simple. In case of half-integer circulation, we show that the rate of
convergence depends on the number of nodal lines of the corresponding magnetic
eigenfunction. In addition, we provide several numerical simulations both on the
circular sector and on the square, which find a perfect theoretical justification within
our main results, together with the ones by the first author and Helffer in
Exp. Math.20:3 (2011), 304–322.
Keywords
magnetic Schrödinger operators, eigenvalues, nodal domains
Département de Mathématiques
Université Libre de Bruxelles (ULB)
Boulevard du triomphe
B-1050 Bruxelles
Belgium
Dipartimento di Matematica e Applicazioni
Università degli Studi di Milano-Bicocca
via Bicocca degli Arcimboldi 8
20126 Milano
Italy