Say $A, B : H \supset D \to H$ are essentially self-adjoint operators on the dense common domain $D$. $H$ is some Hilbert space. Does it hold that $A + B$ is also essentially self-adjoint? If not, can you please give me a counterexample?

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Why don't you just leaf through one of the hundreds of books on spectral theory and/or theory of operators (e.g., Reed and Simon's treatise)? a large part of the theory is devoted to answer precisely this question –  Piero D'Ancona Jun 4 '11 at 8:57

Let $D=H^4(0,1)\cap H^2_0(0,1)$, $Au=u''''$, $Bu=-u''''+u''$.

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It may be helpful to note that the reason counterexamples exist is that the intersection of the two domains may be smaller than each of the domains; and if $A$ is ess. self-adjoint with domain $D$ and $D_0\subset D$ is dense, the closure of the restriction of $A$ to $D_0$ may not coincide with $A$. (If the closure of the restriction of $A$ to $D_0$ is $A$ one sometimes says that $D_0$ is a core for $A$). So your $A+B$ will be ess. self-adjoint if you assume that there is a common core for $A$ and $B$. –  Dima Shlyakhtenko Jun 4 '11 at 7:43
In the example above D is a core for A and B. It is not a core for A+B. –  Michael Renardy Jun 4 '11 at 10:41