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I came across this article [1], in which the author studies some Hamiltonian that have a discrete spectrum even though they do not go to infinity at infinity.

In there, the author makes several claims, that I don't really get :

If $H_1 \geqslant H_2$ and $H_2$ has a discrete spectrum, then so does $H_1$.

If $\operatorname{Tr}e^{-tH}< \infty$ for any $t$, then $H$ has a discrete spectrum.

I would like to understand the intuition behind those results, as well as their formal proofs. Maybe some references on the spectrum of operator on Hilbert spaces could be useful as well.

[1] Some quantum operators with discrete spectrum but classically continuous spectrum; B. Simon (Caltech); Published in: Annals Phys. 146 (1983), 209-220

I came across this article [1], in which the author studies some Hamiltonian that have a discrete spectrum even though they do not go to infinity at infinity.

In there, the author makes several claims, that I don't really get :

If $H_1 \geqslant H_2$ and $H_2$ has a discrete spectrum, then so does $H_1$.

If $\operatorname{Tr}e^{-tH}< \infty$ for any $t$, then $H$ has a discrete spectrum.

I would like to understand the intuition behind those results, as well as their formal proofs.

[1] Some quantum operators with discrete spectrum but classically continuous spectrum; B. Simon (Caltech); Published in: Annals Phys. 146 (1983), 209-220

I came across this article [1], in which the author studies some Hamiltonian that have a discrete spectrum even though they do not go to infinity at infinity.

In there, the author makes several claims, that I don't really get :

If $H_1 \geqslant H_2$ and $H_2$ has a discrete spectrum, then so does $H_1$.

If $\operatorname{Tr}e^{-tH}< \infty$ for all $t$, then $H$ has a discrete spectrum.

I would like to understand the intuition behind those results, as well as their formal proofs. Maybe some references on the spectrum of operator on Hilbert spaces could be useful as well.

[1] Some quantum operators with discrete spectrum but classically continuous spectrum; B. Simon (Caltech); Published in: Annals Phys. 146 (1983), 209-220

I came across this article [1], in which the author studies some Hamiltonian that have a discrete spectrum even though they do not go to infinity at infinity.

In there, the author makes several claims, that I don't really get :

If $H_1 \geqslant H_2$ and $H_2$ has a discrete spectrum, then so does $H_1$.

If $\operatorname{Tr}e^{-tH}< \infty$ for any $t$, then $H$ has a discrete spectrum.

I would like to understand the intuition behind those results, as well as their formal proofs. Maybe some references on the spectrum of operator on Hilbert spaces could be useful as well.

[1] Some quantum operators with discrete spectrum but classically continuous spectrum; B. Simon (Caltech); Published in: Annals Phys. 146 (1983), 209-220

I came across this article, in which the author studies some Hamiltonian that have a discrete spectrum even though they do not go to infinity at infinity.

In there, the author makes several claims, that I don't really get :

If $H_1 \geqslant H_2$ and $H_2$ has a discrete spectrum, then so does $H_1$.

If $\operatorname{Tr}e^{-tH}< \infty$ for all $t$, then $H$ has a discrete spectrum.

I would like to understand the intuition behind those results, as well as their formal proofs. Maybe some references on the spectrum of operator on Hilbert spaces could be useful as well.

I came across this article [1], in which the author studies some Hamiltonian that have a discrete spectrum even though they do not go to infinity at infinity.

In there, the author makes several claims, that I don't really get :

If $H_1 \geqslant H_2$ and $H_2$ has a discrete spectrum, then so does $H_1$.

If $\operatorname{Tr}e^{-tH}< \infty$ for all $t$, then $H$ has a discrete spectrum.

I would like to understand the intuition behind those results, as well as their formal proofs. Maybe some references on the spectrum of operator on Hilbert spaces could be useful as well.

[1] Some quantum operators with discrete spectrum but classically continuous spectrum; B. Simon (Caltech); Published in: Annals Phys. 146 (1983), 209-220