Suppose that $A(t), t \in \mathbb{R}$, is an analytic family of compact selfadjoint operators on a Hilbert space. The KatoRellich theorem says that every nonzero eigenvalue of $A(t)$ splits into one or more analytic eigenvalue functions $\lambda(t)$. These eigenvalue functions can be extended analytically as long as $\lambda(t) \ne 0$. Kato cautions in his book Perturbation Theory of Linear Operators that it might not be possible to continue the eigenvalue function analytically after $\lambda(t) = 0$, but he does not include an example. Are there simple examples where the eigenvalues cannot be extended analytically after reaching 0?
1 Answer
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Consider $$ A(t)=\begin{pmatrix}1 & 1t \\ 1t & t\end{pmatrix}. $$ Then $\lambda_{1,2}(t)=\frac12\left(1+t\pm\sqrt{5}t1\right)$.

$\begingroup$ In this case (and always in finite dimensions) you can relabel the functions before and after the crossing so that the individual functions are analytic. In your example, just let the two functions be $\frac{1}{2}(1+t+\sqrt{5}(t1))$ and $\frac{1}{2}(1+t+\sqrt{5}(t1))$. In infinite dimensions it isn't always possible to do that. In a recent paper of mine https://arxiv.org/abs/2007.03649, I construct an example (see Example 6.2) of this behavior, but my example is complicated. So I still wonder if there are simpler examples. $\endgroup$ Jul 16, 2020 at 17:02