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

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

