Schrödinger operator whose potential analytically depends on a parameter – how does the spectrum change?

Question

Let's say we have a self-adjoint operator $H_s$ on the Hilbert space $L^2(\Omega \subseteq \mathbb{R})$ defined by $$ H_s \, \psi(x) := -\psi''(x) + V_s(x) \, \psi(x) \: , $$ where $s \in \mathbb{R}$ is a free parameter and the real function $V_s(x)$ is analytic in $s$. Let's also suppose that the spectrum of $H_s$ is discrete for every $s$. What can we say about the eigenvalues and their dependence on $s$? Do they change continuously with $s$, or even analytically?

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I've tried searching for a relevant theorem in Kato's Perturbation Theory, but couldn't find it. Any references are welcome!

Answer 1

I think it is in Kato, as theorem VII.3.9 ...

... and, yes, the spectrum depends analytically on the parameter in which the coefficients are (real) analytic.