Suppose $A$ is an unbounded self-adjoint operator in a Hilbert space $H$ with discrete spectrum $$\lambda_0 < \lambda_1 < \cdots$$ bounded below with lowest eigenvalue $\lambda_0$, lowest eigenstate $\psi_0$, and spectral gap $\lambda_1 - \lambda_0 > 0$.
Let $B$ be a bounded self-adjoint operator in $H$ with absolutely continuous spectrum.
Are there any conditions on $A, B$ which imply that the spectral measure $d\mu$ of $O=A+B$ at $\psi_0$ is bounded, i.e. $d\mu$ compact support?
Recall that the spectral measure of a self-adjoint operator $O$ in a Hilbert space $(H, \langle \cdot, \cdot \rangle)$ at a normalized state $\psi \in H$, $||\psi||=1$ is the probability measure $d\mu$ on $\mathbb{R}$ determined by $O$ and $\psi$ implicitly by
$$\int_{- \infty}^{+\infty} e^{\textbf{i} t \lambda} d \mu(\lambda) = \langle \psi , e^{\textbf{i} t O} \psi \rangle $$ for all $t \in \mathbb{R}$.