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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 $\sigma(H)$?

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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.

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  • $\begingroup$ Right. Thanks a lot! But is it true that all the spectra $\sigma (H_k)$ are uniformly bounded from below by a constant? $\endgroup$
    – Onil90
    Commented Jun 2, 2017 at 5:27
  • $\begingroup$ @Onil90 What if V has an infinite number of potential wells of increasing depth? $\endgroup$ Commented Jun 2, 2017 at 15:14

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