Let $A,B$ be two positive unbounded, self-adjoint operators on some Hilbert space that strongly commute. Let $D(A)$ and $D(B)$ denote their respective domain. Then, using for instance the spectral theorem, A+B is self-adjoint on $D(A)∩D(B)$.

If we furthermore assume that $A$ and B are essentially self-adjoint on some common core D, is it also the case of $A+B$ ?