Let $H = -\Delta + V$ be a Schrödinger a self-adjoint operator on $L^2(\mathbb{R}^n)$ with core $C_0^\infty(\mathbb{R^n})$, where $V \in L^2_{loc}(\mathbb{R}^n)$.
Let's define for every $k \in \mathbb{N}$ the following operators: $$ H_k = -\Delta + V \chi_{ \{x\in \mathbb{R}^n : |V(x) | \le n\}} $$
Is it true that the spectrum $\sigma (H_k)$ is contained in the spectrum $\sigma(H)$?
Suppose $V$ is negative and $H$ has a few discrete eigenvalues ("bound states"). You could choose $V$ so that the difference $H - H_k$ is a small perturbation, producing a small shift in the eigenvalues, so they do not coincide.