Let $L: \text{Dom} (L)\subset L^2(\mathbb{R}^3)\mapsto L^2(\mathbb{R}^3)$ be a self-adjoint, closed, and negative operator. I am trying to prove that the discrete spectrum $\sigma_{dis}$ of $L$ can only accumulate at the essential spectrum $\sigma_{ess}$ of $L$.
I am thinking about using only the definition of $\sigma_{dis}$, using the fact that it only includes isolated points. If a sequence $\lambda$ in $\sigma_{dis}$ converges to $\lambda$, then $\lambda$ is not isolated and thus $\lambda$ is in $\sigma_{ess}$.
Is this true on enough? I see in this reference on page 11 proposition 1.1.9 a bit more complicated criteria, that's why I'm doubting about mine.
Thanks for helping me know if I'm mistaken or not!