# Interesting examples of spectral decompositions of BOUNDED operators with both continuous and discrete spectrum

I would like to have a few basic examples of bounded self-adjoint operators $$T$$ (more generally bounded normal would be fine) on a Hilbert space $$(H,\langle,\rangle)$$ for which the following criteria hold true:

(1) Both the discrete and continuous spectra are non-empty

(2) There is a "physical incarnation" of the continuous spectrum by functions $$g_t$$ where $$t$$ is a parameter which varies over the continuous spectrum $$\sigma_c(T)$$ of $$T$$.

(3) We have an explicit spectral decomposition of $$T$$, i.e., if $$\{u_i: i\in I\}$$ is a complete set of eigenvectors for the discrete spectrum of $$T$$ then for any function $$f\in H$$ we have an equality of the following type: $$f=\sum_{i\in I}\langle f,u_i\rangle u_i+ C\int_{t\in\sigma_c(T)} \langle f,g_t\rangle g_t\cdot\mu(t),$$ where $$C$$ is a suitable constant and where $$\mu(t)$$ is an appropriate measure on $$\sigma_c(T)$$.

As a number theorist the only explicit examples that I know of of such spectral decompositions is when one deals with Laplacian-like differential operators, say $$\Delta$$, acting on a suitable Hilbert space associated to a space of automorphic forms on a non-compact locally symmetric space. Note though that for these examples $$\Delta$$ is never a bounded operator. Moreover, in this case, the "physical incarnation" of the continuous spectrum is provided by explicit Eisenstein series which barely fail to be square integrable, so in sense, I like to think of these Eisenstein series as living in a "neighbourhood of H".

In any case, I would be quite happy to have a few basic examples at my disposal of BOUNDED operators satisfying (1), (2) and (3).

added By "physical incarnation" I mean that $$T$$ can still be applied to $$g_t$$ (even if $$g_t$$ fails to be in the domain of $$g_t$$) and $$T g_t=tg_t$$. I guess that in practice there could be a discrete set of $$g_t^{i}$$ such that $$Tg_t^i=tg_t^i$$. For example if one considers the usual Laplacian $$\Delta=-d/dx^2$$ on $$\mathbb{R}$$ then for $$\xi\in\mathbb{R}$$, $$[x\mapsto e^{ix\xi}]$$ and $$[x\mapsto e^{-ix\xi}]$$ (two functions not in the domain of $$\Delta$$) both have eigenvalues $$\xi^2\in\sigma_c(\Delta)$$.

• If $T$ is a self-adjoint operator then $(i Id - T)^{-1}$ is a bounded normal operator, whose spectral decomposition can be deduced from the decomposition of $T$. So boundedness is not really an issue. – Mateusz Wasilewski Dec 6 '18 at 16:33
• Yeah, what's the definition of "physical incarnation"? I'm trying to think about the most basic example of a bounded operator with continuous spectrum: multiplication by $x$ on $L^2([0,1], dx)$. I can't figure out whether it satisfies your criteria or not. – Nate Eldredge Dec 6 '18 at 18:01
• @NateEldredge: Your example doesn't have discrete spectrum, but of course that's easily fixed by adding a discrete part $\sum \delta_{x_j}$, with $x_j\notin [0,1]$, to $dx$. – Christian Remling Dec 6 '18 at 18:39
• Well I like when I can touch and compute things. For example Eisenstein series admit Fourier series expansion and this gives you a good hold on these objects. I read a little bit about the spectral theorem for unbounded self-adjoint operators and I did not get much insight regarding the associated spectral measure. At the end I would like to be able for any f in H to write it as a weighted sum of explicit eigenvectors for which I have a good hold. – Hugo Chapdelaine Dec 6 '18 at 20:43
• @HugoChapdelaine: Yes, such generalized eigenfunctions (if we want to call them that) can be introduced in many situations, but I don't think they make anything clearer. Quite on the contrary, they obscure what is going on, and the rigorous version of the spectral theorem gives a much clearer picture. – Christian Remling Dec 6 '18 at 23:58