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Suppose that $A(t), t \in \mathbb{R}$, is an analytic family of compact self-adjoint operators on a Hilbert space. The Kato-Rellich theorem says that every non-zero 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?

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Consider $$ A(t)=\begin{pmatrix}1 & 1-t \\ 1-t & t\end{pmatrix}. $$ Then $\lambda_{1,2}(t)=\frac12\left(1+t\pm\sqrt{5}|t-1|\right)$.

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  • $\begingroup$ In this case (and always in finite dimensions) you can re-label the functions before and after the crossing so that the individual functions are analytic. In your example, just let the two functions be $\frac{1}{2}(1+t+\sqrt{5}(t-1))$ and $\frac{1}{2}(1+t+\sqrt{5}(t-1))$. 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. $\endgroup$
    – Brian Lins
    Jul 16, 2020 at 17:02

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