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I am using the "Direct integral" version of the spectral theorem, as given e.g. in Hall. It says that a diagonal representation for an unbounded operator on the Hilbert space $L^2(\mathbb{R}^n)$ can be constructed as a "direct integral", or

$$ \mathcal{H}_{\hat{A}} = \int^\oplus_{\sigma(\hat{A})}d\mu(a) \mathcal{H}_a. $$

States in the Hilbert space are formed as square-integrable "sections", or as functions of $a$ with values in the Hilbert spaces $\mathcal{H}_a$. Since the $\mathcal{H}_a$ spaces are not true subspaces, vectors in them are not actually in $\mathcal{H}$, so are a "generalised eigenvectors". They can be defined as a functional on a dense subset of the Hilbert space by means of

$$ v \in \mathcal{H}_a: \; \psi(a^\prime) \; \mapsto (\psi(a),v) \in \mathbb{C}, $$

which can then of course be extended back to functionals on $L^2(\mathbb{R}^n)$. I'm fairly sure that this chapter's results can be used to demonstrate that if $\hat{A}$ is in the algebra generated by $\hat{x}$ and $\hat{p}$ then these functionals will necessarily be tempered distributions.

Now, the specific $\hat{A}$ I'm interested in is a differential operator on $L^2(\mathbb{R}^{3\oplus1})$. (Specifically, it's the "Klein-Gordon-like" operator used for spin-half particles in an applied electromagnetic field, formed by squaring the Dirac operator.) What I would really like to be able to do is, given that $\hat{A}$ is a differential operator, use some results about PDEs on these generalised eigenvectors.

This, though, requires some notions about the regularity of the generalized eigenfunctions, which I have just not been able to find. My issue is that the operator $\hat{A}$ I'm using is second-order Hyperbolic, and so there aren't many good results about regularity in the PDE maths literature for me. This seems to be because the mere fact that $\psi$ obeys the differential equation $(\hat{A}-a)\psi=0$ isn't enough to enforce any sort of regularity, for a similar reason that $\delta(x-ct)$ is a solution to the wave equation. I think that I therefore need regularity enforced by the use of the solution to the differential equation as a generalized eigenfunction, or as a basis state in the direct integral spectral representation.

I realise this might be a bit of a long shot, but I've struggled for days to find such a result, and it would be extremely useful for me. Does anyone know of any literature which deals with this question of the regularity of the basis of spectral representations of differential operators?

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  • $\begingroup$ Actually, there is a version of set theory adapted for measure theory where all linear operators are bounded; this might make life much easier... $\endgroup$ Commented Mar 18, 2021 at 0:39
  • $\begingroup$ Minor comment to the post (v2): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. $\endgroup$
    – Qmechanic
    Commented Mar 18, 2021 at 3:57

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This isn't a direct result about the regularity of generalised eigen-functions, but it is indirectly related:

A dream model (for analysis) is a model of ZF+DC and where every subset of the reals

  1. Lesbegue: Is a union of a Borel set and a null set
  2. Perfect: It is the union of a perfect set and countable set
  3. Baire: It is the union of an open and a meager set.

To cut to the chase: it turns out that in the dream model every total linear map in Banach spaces is bounded! Moreover, any two norms are topologically equivalent. This latter property is true in finite dimensions in ordinary analysis, but dream analysis makes it true for infinite dimensions. This dispells one of the troubles that plagues infinite dimensional analysis: the superabundance of inequivalent norms on infinite dimensional spaces.

(Note that a set in a topological space is perfect when it is closed and has no isolated points. Notably, each of these axioms is contradicted by Choice. Thus we see there is life in mathematics (and in physics) without Choice (but we do have DC - dependent choice). Moreover, Solavay showed that a dream model is consistent if an inaccessible cardinal is consistent with ZF. For category theoretic purposes we often already assume the existence of an inaccessible cardinal, this in other terms, is the so-called Grothendieck universe hypothesis).

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