Both the fundamental examples of Schrodinger Operators (Harmonic Oscillator, Hydrogen Atom ones) and the physical intuition suggests that the discrete spectrum of $-\Delta+V$ is always a subset of the Image of $V$. Is this true? And how is it proven? Is it linked with the spectrum of the multiplication operator by $V$ being the aforementioned image? Referencies are much appreciated. Thank you.
Edit: an answer showed me this is generally false, but the example is not on the whole $\mathbb{R}^{d}$ . I am interested in knowing if this is true in that case. I apologize for having omitted that detail.
Edit2: this question is stupid. I keep asking the wrong thing (and the above stated questions are easily answered as false). What I want to know is whether the discrete spectrum of $-\Delta+V$ on the whole $\mathbb{R}^d$ is a subset of $[\inf{V}, \sup{V}]$