Scattering, determinants, hyperfunctions in relation to $\Gamma(1-s)/\Gamma(s)$
Open Letter
I wrote this paper in 2006, and submitted it to a journal specializing in integral equations and operator theory. After circa 14 months I received a report which I reproduce in full here (I allow myself to correct the spelling of a mathematician's name cited in the report):
"In spite of desperate efforts, the referee has failed to understand what the paper is about. Apparently it does not have a definite goal but consists of miscellaneous remarks to the papers by de~Branges and Rovnyak. It is practically impossible to distinguish original results in this jumble. Actually, the text does not look as a mathematical article but rather as some notes for personal use.
In the referee's opinion, the paper should be rewritten according to conventional rules and its volume should be divided by the factor 5-10. The author should try to formulate the results which he considers to be new."
Let me explain why I consider the publication of the paper important. First of all the referee's report only serves to demonstrate that the referee did not read the manuscript. I tried to point this out to the editor in chief, to no avail:
"Dear Professor Burnol,
I read all your letters to us. I am not changing my mind! Your paper is not accepted for publication. This decision is final and the discussions about this paper this time I consider finished.
Sincerely, XXX"
I think this illustrates nicely how dysfunctional the peer-review process may be, at times. Regarding the paper itself, it is well structured, and its goal was to prove new mathematical theorems (!), a goal which was achieved (!). I corrected a typo in 2008 (there was a superfluous imaginary i in some equations, see the footnote on page 1), this is the only change to the 2006 version.
The referee asked me to divide the "volume" by between five and ten, a request which at that time particularly infuriated me. In fact, a more acceptable comment would have been to point out that the paper contained material for between 3 and 5 reasonably sized quasi-independent publications (of reasonable, but obviously not earth-shaking interest!), but I wanted to make a common exposition with in particular a common introduction. What would be the point of repeating 5 times the same introduction? An introduction is made necessary by the fact that my perspective is unique and links together a priori disjoint topics, the reader needs some help in entering this framework.
Another difficulty is that in 2008, during a stay at Institut des Hautes Études Scientifiques (IHES), I made very significant advances (establishing links with domains apparently completely unrelated, and which moreover have been of great interest for the last thirty years to large communities of researchers), on which I have had opportunities to give lectures at IHES, at the European Conference of Mathematics (ECM) at Amsterdam, and at a workshop at the Independent University of Moscow (Conference Zeta functions II). I have circulated a hand-written manuscript of about 80 pages, and prior to publishing this novel material in peer-reviewed journals, I need to make my earlier work available to the mathematical community.
I did sufficiently serious and dedicated work on this in 2006 resulting in a paper of about 65 pages. It would be all too easy, and far more beneficial to my career, to instead divide the paper into at least 3 publications, but I just don't see the point. If one is not sufficiently committed to mathematics to place great importance on the form one gives to one's own contributions, if one is ready to obey arbitrary diktats, if all that matters is adding lines of publications to a CV, then one practices a job and not a passion and one does not care about his/her legacy, one lives amidst superficial illusions and pleasures.
This paper will be necessary reading to get a full understanding of my earlier as well as of my future works.
Abstract
The method of realizing certain self-reciprocal transforms as (absolute) scattering, previously presented in summarized form in the case of the Fourier cosine and sine transforms, is here applied to the self-reciprocal transform $f(y) \mapsto H(f)(x) = \int_0^\infty J_0(2\sqrt{xy})f(y)\,dy$, which is isometrically equivalent to the Hankel transform of order zero and is related to the functional equations of the Dedekind zeta functions of imaginary quadratic fields. This also allows to re-prove and to extend theorems of de Branges and V. Rovnyak regarding square integrable functions which are self-or-skew reciprocal under the Hankel transform of order zero. Related integral formulae involving various Bessel functions are all established internally to the method. Fredholm determinants of the kernel $J_0(2\sqrt{xy})$ restricted to finite intervals $(0,a)$ give the coefficients of first and second order differential equations whose associated scattering is (isometrically) the self-reciprocal transform $H$, closely related to the function $\frac{\Gamma(1-s)}{\Gamma(s)}$. Remarkable distributions involved in this analysis are seen to have most natural expressions as (difference of) boundary values (i.e., hyperfunctions). The present work is completely independent from the previous study by the author on the same transform $H$, which centered around the Klein-Gordon equation and relativistic causality. In an appendix, we make a simple-minded observation regarding the resolvent of the Dirichlet kernel as a Hilbert space reproducing kernel.
