In the article "Eigenvalues variations for Aharonov-Bohm operators", Corentin Léna proves local analyticity of the eigenvalues of Aharonov-Bohm operators with respect to the poles with two restrictions:
- That we are in the neighborhood of a pole configuration where the eigenvalue is simple;
- That we are in the neighborhood of a pole configuration where no pole is on $\partial\Omega$.
Now I understand the reason for restriction 1: if the eigenvalue is multiple, we conclude the eigenvalues are branches of analytic functions, but no-one guarantees that the analytic functions are the eigenvalue functions, maybe they branch in a different way, i.e. there is an analytic function which coincides with one eigenvalue under some conditions and with another one under others. But why restriction 2? This cannot, AFAICT, give problems when applying the Katō-Rellich perturbation theory, since that theory is independent of the poles, so the problem must be in the analyticity of the family of $r_t$. But I cannot see how that restriction enters the proof of this analyticity. Any ideas?
If you look at the proof, at the very beginning, Léna constructs the $\Phi_t$, which he the states to be diffeomorphisms $\Omega\to\Omega$ for $t$ small enough. If any ball meets the boundary, it is certain that, when translating the small balls by $(t_{2i-1},t_{2i})$, either the part of it in $\Omega$ ends up partially outside $\Omega$ (thus breaking $\Phi_t(\Omega)\subseteq\Omega$), or the part outside ends up partially inside (thus breaking the reverse inclusion). Both cases impede $\Phi_t(\Omega)=\Omega$, and hence prevent $\Phi_t$ from being a diffeomorphism $\Omega\to\Omega$, a condition that is used to prove unitariness of $U(t)$, which is needed for the proof to work. Thus, the condition is necessary.