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The Kato-Rellich Theorem states: If $T$ is a self-adjoint operator in the hilbert space $\mathcal{H}$ and if $A$ is symmetric and $T-\text{bounded}$ with $T$-bound $< 1$. Then $T+A$ is self-adjoint. I´m wondering if the following is true:

$$\text{If}~~T~~\text{is s.a. in}~~\mathcal{H}~~\text{and if}~~A~~\text{is symmetric and if it holds that}~~\overline{T+A} = \left(T+A\right)^{*},~~\text{then}~~A~~\text{is}~~T-\text{bounded}.$$

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A counterexample is the quartic quantum anharmonic oscillator $H+V$ with $V=x^4$. On the Hermite function basis $\{e_n\}$, $He_n = ne_n$ while $\vert Ve_n \vert \gt$ const.$\times n^2$. The latter is obvious from the creation/annihilation representation. There are various proofs of e.s.a. for $H+V$.

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