Papers by D. Carfi: extended spectral theory of distributions.

[strangerep]

Over in the Quantum Physics forums, we occasionally have threads involving rigged Hilbert space -- a.k.a. Gel'fand triple: $\Omega \subset H \subset \Omega'$ where $H$ is a Hilbert space, $\Omega$ a dense subspace thereof such that certain unbounded continuous-spectrum operators are well-defined everywhere thereon, and $\Omega'$ is its topological dual. A recent thread of this kind is: https://www.physicsforums.com/showthread.php?t=668013

With certain extensions of the meaning of "self-adjoint operator" to the space $\Omega'$, some treatments of QM rely on the so-called nuclear spectral theorem (cf. Gelfand & Vilenkin vol 4) which basically assures us that the generalized eigenvectors of a self-adjoint operator $A$ in $\Omega'$ span $\Omega$, and that operators of the form $f(A)$, for analytic functions $f$, make sense. This is basically a generalization of the usual spectral theorem for unbounded operators on Hilbert space to the distributional context of rigged Hilbert space.

I've always felt it unsatisfactory that arbitrary elements of the dual space $\Omega'$ are not also covered by such a theorem. Recently, David Carfi put a series of papers on the arXiv claiming to do just this (and he confirmed to me in brief private correspondence that this is indeed his intent). In reverse time order the papers are:

http://arxiv.org/abs/1104.4660
http://arxiv.org/abs/1104.4651
http://arxiv.org/abs/1104.3908
http://arxiv.org/abs/1104.3380
http://arxiv.org/abs/1104.3324
http://arxiv.org/abs/1104.3647

Unfortunately, my abilities in Functional Analysis, etc, are inadequate to form a reliable opinion about these papers. They have not been published in peer-reviewed journals (afaict), but David Carfi seems to have published elsewhere in financial and economics mathematics.

So... I'm hoping that the FA experts here can spare a little time to look through Carfi's papers and figure out whether he does indeed achieve what I described above, i.e., establish a sensible spectral expansion for all elements of the distribution space $\Omega'$, and not merely the well-behaved elements from $\Omega$.

 

[micromass]

I got to admit that these papers look very interesting. I think I'm going to waste the coming days trying to work through them.

If you take a look at the last paper: http://arxiv.org/abs/1104.3647 The Theorem on page 3 really does seem something you want. But that's on a very first glance.

I'll read through the papers and post my impressions here. Maybe it would be nice if you could do the same!

 

[strangerep]

I'll read through the papers and post my impressions here.

Thank you! Of course, I was kinda hoping all along that you would!

Maybe it would be nice if you could do the same!

I already looked through them when they first appeared on the arXiv -- that was why I emailed David to ask him a couple of questions. But... I had to admit defeat, due to my inadequate FA abilities. (You were not active in the QM forums back then.)

I will, of course, now try again...

 

[micromass]

So, I've read the first paper: http://arxiv.org/abs/1104.4651

The paper introduces families of distributions. So let $\mathcal{S}_n^\prime$ be the dual of the Schwartz function (thus the tempered distributions). We are interested in functions $v:I\rightarrow \mathcal{S}_n^\prime$.

This notion can be used to generalize quite some good things. Most important, there is a bijective correspondence between such functions and operators between Schwartz classes. This is easy to see in finite dimensions. Let $A$ be a linear map from $\mathbb{R}^n$ to $\mathbb{R}^m$. Then we can of course express $A$ as a matrix. Now we can look at

$$\{1,...,m\}\rightarrow (\mathbb{R}^m)^*: i\rightarrow \text{Row}_i$$

So we have a family that sends a number i to the i'th row. Of course, the rows of a matrix are easily seen to be functionals.

We want to generalize this notion to infinite-dimensional spaces. So we want to talk about rows of infinitary matrices. But for this, we need to find some kind of "basis" for the Schwartz class. Now, it turns out that a basis for [ite]\mathcal{S}_n[/itex] is given by $\delta_x,~x\in \mathbb{R}^n$, the Dirac delta function. Using this, we can now construct an isomorphism
$$\mathcal{B}(\mathcal{S}_n,\mathcal{S}_m) \leftrightarrow \mathcal{S}(\mathbb{R}^n,\mathcal{S}_m^\prime)$$
The set on the left side are the bounded functions between n-dimensional and m-dimensional Schwartz class. The set on the right side are the families $\mathbb{R}^n\rightarrow \mathcal{S}_m^\prime$ which satisfy some continuity property.

Then there is a notion of "summable". I think it is based on the following. Consider a fixed real sequence $v=(x_n)_n$. We can consider this to be "summable" if the series $\sum_n x_n$ exists.
Now, let's define some functions. Let $A\subseteq \mathbb{N}$ finite. Define
$$\hat{v}(A) = \{(y_n)_n ~\vert~ y_n=x_n~\text{if}~n\in A~ \text{and otherwise}~y_n=0\}$$
Then define
$$a(\hat{v}(A)) = \sum y_n = \sum_{n\in A} x_n$$
This is a finite sum. Now, we can define $(x_n)_n$ to be summable if the function
$$u(A) = a(\hat{v}(A))$$
is a convergent net.

Now, in the continuous case, we have a family $v$ of Schwartz functions instead of a sequence. But the definition remains quite the same thing.

The paper then gives some criteria of summability and some equivalence notions. For example, he gives criteria with which we can detect summability of $v$ for more "algebraic" properties such as the existence of a transpose.

I think most of these notions can be generalized a great deal. The paper only shows things for the Schwartz class, but I'm sure they hold more generally too. Maybe we can show versions of arbitrary nuclear spaces? I think that his other papers do things more generally though...

 

[strangerep]

If you take a look at the last paper: http://arxiv.org/abs/1104.3647 The Theorem on page 3 really does seem something you want. But that's on a very first glance.

I've just read through that paper again. But... either I don't properly understand what he's doing, or something important is missing.

The statement of the theorem of $^S$spectral expansion of an operator $A\in L(S_n')$ seems like it only covers the tempered distributions $u$ in the $^S$linear hull $^S$span$(v)$ of the eigenfamily $v$.

But this seems a bit trivial to me. Ordinary spectral theorems show that any vector can be expanded in terms of the eigenvectors of a self-adjoint operator, and that (almost) any other operator can be expanded in terms of those eigenvectors. This seems rather more general (and more difficult) than what Carfi is doing in arXiv:1104.3647.

So, I've read the first paper: http://arxiv.org/abs/1104.4651
[...explanation...]

Thanks. Your explanation allows me to follow that paper much better.

However, I still struggle to understand the connection (in any) between the way he "expands" distributions in terms of Dirac deltas, and the spectrum of a self-adjoint operator. He talks about transposability, but that's just a way of extending the operator from $S$ to $S'$, isn't it? To get a genuine spectral theorem more powerful than the usual nuclear spectral theorem, one needs more than that, or so I would have thought.
We need to express the expansion of any tempered distribution, or operator on $S'$, in terms of the generalized eigenvectors of any self-adjoint operator (or some generalization or modification thereof).

 

[micromass]

I've just read through that paper again. But... either I don't properly understand what he's doing, or something important is missing.

The statement of the theorem of $^S$spectral expansion of an operator $A\in L(S_n')$ seems like it only covers the tempered distributions $u$ in the $^S$linear hull $^S$span$(v)$ of the eigenfamily $v$.

But this seems a bit trivial to me. Ordinary spectral theorems show that any vector can be expanded in terms of the eigenvectors of a self-adjoint operator, and that (almost) any other operator can be expanded in terms of those eigenvectors. This seems rather more general (and more difficult) than what Carfi is doing in arXiv:1104.3647.

Yes. But he never demands self-adjoint anywhere. He has given an integral representation of arbitrary operators. Somewhere, there should be a theorem that the $^S$-span of a self-adjoint operator is the entire space.

 

[strangerep]

[...] he never demands self-adjoint anywhere. He has given an integral representation of arbitrary operators. Somewhere, there should be a theorem that the $^S$-span of a self-adjoint operator is the entire space.

I've read through the other papers again, but I didn't notice anything like that.

No doubt you read the papers with more insight than me, but if you also don't find something like that soon, then we should probably just forget about it.

 

[dextercioby]

Just a personal note: I think these articles are highly irrelevant to a (mathematical) physicist and it only remains to assert their value for the field of pure mathematics.