The Kato-Rellich Theorem is a classical result stating that if $A,B$ are unbounded operators on a Hilbert space with $A$ self-adjoint, $B$ symmetric, $\mathcal D (A)\subset \mathcal D(B)$ and $$ \|Bx\|\leq a\|A x\|+b\|x\|,\;\;x\in\mathcal D(A) $$ for positive constants $a,b$, with $a<1$, then $A+B$ is self-adjoint on the domain of $A$. The infimum of those $a$ for which the above inequality holds is called the relative bound of $B$. Thus, $A+B$ is self-adjoint if $B$ is relatively bounded with bound $a<1$.
I'm interested in non-examples where $A+B$ is not essentially self-adjoint even though $A$ is self-adjoint, $B$ is symmetric and $\mathcal D(A)\subset\mathcal D(B)$.
I have tried considering multiplication operators to no avail. Let $M_\phi$ denote the multiplication operator $$ M_\phi h=\phi h,\;\;\mathcal D(M_{\phi})=\{ h|\phi h\in L_2\}, $$ and let $A=M_f$, $B=M_g$ with $f,g$ real functions, finite almost everywhere. Since $A$ and $B$ are closed, the assumption $\mathcal D(A)\subset\mathcal D(B)$ implies the bound $$ \|B h\|\leq C(\|A h\| +\|h\|) $$ for some $C>0$. But then, for $h$ non-zero only on the set $\{x||f(x)|\leq M\}$, one obtains $$ \|B h\|\leq C(1+M)\|h\|, $$ which implies that $|g(x)|\leq C(1+M)$ almost everywhere on $\{x||f(x)|\leq M\}$. This in turn implies $|f(x)+g(x)|\leq (M+ CM + C)$ on that same set. But then $\mathcal D(A)$ is a core for $\mathcal D(M_{f+g})$, which implies that $A+B$ is essentially self-adjoint.
Next up were Schrödinger operators. 'Unfortunately', Theorem XIII.96 in Methods of Modern Mathematical Physics vol 4 by Reed and Simon implies that (on $\mathbb R^d$, with $d\leq 3$) the inclusion $\mathcal D(-\Delta)\subset\mathcal D(V)$ is sufficient in order to conclude that $V$ is relatively $(-\Delta)$-bounded with bound $0$.
