Relating different definitions of spectra

I am currently reading the paper by Deconinck and Kutz [Computing spectra of linear operators using the Floquet–Fourier–Hill method, J. Comp. Phys., 2006] and I am struggling to understand the relation between what they are calling spectrum and the classical spectral theory.

Following their ideas, consider an operator $$L = \sum\limits_{k=0}^M f_k(x) \partial_x^k,$$ where $$f_k(x)$$ are periodic (and can even be vectors), and define the spectrum $$\sigma(L)$$ as $$\sigma(L) = \{\lambda \in \mathbb{C}; \Vert w \Vert<\infty\},$$ where $$w$$ satisfies $$Lw=\lambda w$$ and is in a function space of interest. In this case, they use the space of bounded functions and the corresponding norm.

My question is: what is the relation between this spectrum and the classical spectra that comes with Functional Analysis? And here you can take any definition you want (point/continuous/residual or essential/discrete, etc), I simply cannot find a relation with any of them.

I ask this because in the same paper the authors say

"Depending on the system under consideration, some basic properties and bounds on the spectrum may be known. This is the case for self-adjoint systems (the spectrum is confined to the real line), Hamiltonian systems (eigenvalues occur in quadruplets), etc"

which means they are using the classical theory of closed, densely defined operators. Usually (other papers by Deconinck), these operators are being considered with domains $$H^1(\mathbb{R}), H^2(\mathbb{R})$$ in the Hilbert space $$L^2(\mathbb{R})$$.

Updated question: I am going to reformulate the question as I seem to understand the motivation behind it.

I started reading about quantum mechanics and how they needed to generalize the concept of eigenvector (or eigenket) for some reason that I am too ignorant to understand. And then we end up with the notion of rigged Hilbert spaces and a whole 'generalised' spectral theory.

So consider a closed densely defined unbounded operator $$L$$ in $$L^2(\mathbb{R})$$ and the 'spectrum' $$\sigma(L) = \{\lambda \in \mathbb{C}; Lw = \lambda w\,\ \text{for}\, w\, \text{bounded}\}.$$

Considering this context of quantum mechanics and the necessity of extending the Hilbert space, I have the following questions:

1. Does it make sense to look at the space of bounded functions in some sort of rigged Hilbert space? The only concrete example I can find is the Schwartz space.
2. Am I to understand that this is indeed the point spectrum, but sort of generalised? In this case, why doesn't the extension of the space affect self-adjoint operators and their real spectrum?
3. In case I am still clueless about everything (very possible), is there any light to be shed here?
• This is a definition of point spectrum. Commented Oct 13, 2022 at 16:13
• But for a $\lambda$ to be in the point spectrum, the eigenvectors need to be in the Hilbert space, right? This is generally not the case with this definition. The spectrum here requires the eigenvectors to be only bounded and not square integrable (assuming the Hilbert space to be $L^2(\mathbb{R})$). Commented Oct 13, 2022 at 18:59
• You don't need a Hilbert space to define point spectrum. You can define point spectrum for any operator on any vector space; you don't even need a norm or a topology, since you only need the vector space structure to define eigenvectors and eigenvalues. Commented Oct 13, 2022 at 19:23
• I do need it to define self-adjointness, something that is mentioned in the paper and that is constantly used in the literature. Or maybe not, I am very confused right now. Commented Oct 13, 2022 at 19:34
• Yes, admittedly I have no idea how this definition interacts with usual spectral theory on Hilbert spaces. Do the authors say explicitly that they want to consider eigenvectors not living in $L^2$? Commented Oct 13, 2022 at 19:37