Spectral stability under removal of small capacity sets and applications to Aharonov–Bohm operators

Abstract

We first establish a sharp relation between the order of vanishing of a Dirichlet eigenfunction at a point and the leading term of the asymptotic expansion of the Dirichlet eigenvalue variation, as a removed compact set concentrates at that point. Then we apply this spectral stability result to the study of the asymptotic behaviour of eigenvalues of Aharonov-Bohm operators with two colliding poles moving on an axis of symmetry of the domain.

On the multiplicity of the second eigenvalue of the Laplacian in non simply connected domains – with some numerics –

“…More details (which are not necessary for the argument) will be given in Section 7. Here we see from (5.1) that λ 2 (B R2 ) has multiplicity two corresponding to σ(H (1)…”

“…We obtain a similar expansion for the other eigenvalue, starting from Theorem 1.4 in [1], which gives us…”

“…Since u is radially symmetric, u(x + ) = u(x − ). We then observe that the proof of Proposition 1.5 in [1] can be adapted to give…”

“…The family of compact sets (K δ ) δ>0 concentrates to the set {x + , x − }, in the sense that K δ is contained in any open neighborhood of {x + , x − } for δ small enough. Reference [1] provides two-term asymptotic expansions in this situation. A direct application of Theorem 1.7 in [1] gives…”

“…Reference [1] provides two-term asymptotic expansions in this situation. A direct application of Theorem 1.7 in [1] gives…”