I am trying to form intuition for the following `well-known' facts about spectrum of unbounded operators (Schrodinger/wave etc.) $L$ on $\mathbb{R}^n$.

Let $\lambda\in\mathbb{R}$ satisfy $Lf=\lambda f$, for some function $f$.

Roughly speaking: we call "eigenvalues" the things that give us $f$ in some $L_2$ space. If the eigenvalue equation is satisfied but the $f$ is not in $L_2$ (roughly, it doesn't decay at infinity), then we don't call $\lambda$ an eigenvalue, but consider it part of essential (or continuous ?) spectrum. Example of latter is the usual Laplacian.

Is there a resource where I can understand how generic is this "equivalence":

Existence of "bound state" <-> Existence of eigenvalue

I am trying to form intuition for the following `well-known' facts about spectrum of unbounded operators (Schrodinger/wave etc.) $L$ on $\mathbb{R}^n$.

Let $\lambda\in\mathbb{R}$ satisfy $Lf=\lambda f$, for some function $f$.

Roughly speaking: we call "eigenvalues" the things that give us $f$ in some $L_2$ space. If the eigenvalue equation is satisfied but the $f$ is not in $L_2$ (roughly, it doesn't decay at infinity), then we don't call $\lambda$ an eigenvalue, but consider it part of essential (or continuous ?) spectrum. Example of latter is the usual Laplacian.

Is there a resource where I can understand how generic is this "equivalence":

Existence of "bound state" <-> Existence of eigenvalue

I was motivated to ask this after reading the following : https://en.m.wikipedia.org/wiki/Bound_state_in_the_continuum