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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 $\xi^2\in\sigma_c(\Delta)$.

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    $\begingroup$ 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. $\endgroup$
    – Mateusz Wasilewski
    Commented Dec 6, 2018 at 16:33
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    $\begingroup$ 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. $\endgroup$
    – Nate Eldredge
    Commented Dec 6, 2018 at 18:01
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    $\begingroup$ @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 $dx$. $\endgroup$
    – Christian Remling
    Commented Dec 6, 2018 at 18:39
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    $\begingroup$ 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. $\endgroup$
    – Hugo Chapdelaine
    Commented Dec 6, 2018 at 20:43
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    $\begingroup$ @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. $\endgroup$
    – Christian Remling
    Commented Dec 6, 2018 at 23:58

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