Analyticity of eigenvalues: results from Katō and real-analytic forms

Question

In the article "Eigenvalues variations for Aharonov-Bohm operators", Corentin Léna constructs a family of forms $r_t$ for $t\in\mathbb{R}^{2N}$ and proves that, for $u$ in a suitable Banach space (form domain of an Aharonov-Bohm operator) and for $t$ small enough, $r_t(u)$ is analytic w.r.t. $t$. He then uses Kato-Rellich theory to conclude the eigenvalues of these forms are analytic for $t$ small enough. Now, assuming said eigenvalues are those of Aharonov-Bohm operators (which is a whole other problem), I have a problem with this step. I looked at the reference given in Léna, which is Toshio Katō, Perturbation Theory for Linear Operators, and the results therein are about holomorphic families of forms, so the parameter is a single complex variable and the map $\lambda\mapsto r_\lambda(u)$ is holomorphic. In the article, AFAICT, $t\mapsto r_t(u)$ is only real-analytic, not complex-analytic, and $t$ is in $\mathbb{R}^{2N}$. Now, the latter problem may be solved by interpreting $t$ as $\mathbb C^N$ the usual way. However, holomorphy is still required to use the book's proof of analytic forms imply analytic operators, because it uses the Cauchy integral formula. Moreover, the step to analytic eigenvalues depends on "a well-known result", and I don't know whether it generalizes to multiple complex parameters or to real-analytic multi-variable contexts. So I was wondering: is there another version of that book with more relaxed hypotheses that he was looking at? Or is there another reference that extends that book's results to these more relaxed hypotheses? Or how can such an extension be carried out?

Answer 1

Real-analytic means it's a real power series, so if you extend it to complex numbers in the obvious way you will have a complex power series, which is complex-analytic aka holomorphic, so you can apply Kato to that, obtain the holomorphicity of the eigenvalues of the extension, and then restrict to real numbers once more, obtaining the real-analyticity of the eigenvalues, as desired.