In 2021, Alain Connes and Henri Moscovici discovered a new spectrum for the prolate spheroidal operator of order zero 
 \[
 W_\Lambda=-\frac{d}{dx}(\Lambda^2 – x^2)\frac{d}{dx} +(2\Lambda x)^2. 
 \]
 This CM-spectrum is the spectrum of a self-adjoint extension introduced by A. Connes in 1998. It is discrete, contains a replica of the classical spectrum (which is positive) and an infinitely many negative eigenvalues.
 CM observed that, for $\lambda=\sqrt 2$, their (even) spectrum “matches the zeroes” of zeta. Unfortunately, we made no progress on this (essential …) point. Our initial aim was to give a rather complete study of the CM-spectrum, extending the numerous known result for the classical spectrum, but we got some completely new (surprising) properties of the non classical CM-spectrum. 
 
There is nothing new on the “arithmetic properties” of the spectrums except a strange result that I will describe at the end of my lecture. 

The CM approach is based on classical self-adjoint operators theory. Ours is very different. It is based on the structure of the singularities of a rational linear differential equation in the complex domain and analytic continuation. As I will explain at the end of my lecture, these datas can be organized into a notion of wild monodromy representation which is a geometric counterpart of the equation in Riemann-Hilbert line. 
   
1.  Connes-Moscovici work.  

Firstly I will recall briefly some results of CM.   


2. Complex analytic tools. 

I will recall basics on singularities of a rational linear differential equations of order two in the complex domain: regular-singular points, irregular points, monodromy, Borel-summability of divergent power series solutions and Stokes phenomenum. I will detail the case of $W_\Lambda$.
 
   
3. Analytic interpretations of the CM spectrum. Application to numerical computations. 

I will compare different approaches of the spectrums: naive, Sturm-Liouville, analytic. The idea of the last one is to select a line of “distinguished solutions” at two singular points using only the singularity theory and to compare these two lines using analytic continuation along a simple continuous path joining the points. 
 
 The analytic approach gives `functional determinants” whose zeros are the eigenvalues. They are entire functions of order less than 1/2. This approach also allowes a rather efficient numerical computation of the eigenvalues. 
 
4. New eigenfunctions associated to a spectrum along the imaginary axis.

We discovered new eigenfunctions for the non-trivial part of the CM-spectrum. They appear in the study of (simple) spectral problems on the imaginary axis. The CM eigenfunctions are the  boundary-values of the analytic extension of these new eigenfunctions over a cut plane.
 
Using this new interpretation of the non classical part of the CM-spectrum, we prove that all the non trivial CM-eigenvalues are negative and we give a very efficient method of numerical computation of these eigenvalues. 
 

5. Extension to $\Lambda \in \mathbb{C}^*$ and analytic continuation of the eigenvalues. 

In their monograph, in 1955, J. Meixner and F.W. Scha\”fke gave a remarkable study of the complex version of the spherical operators and of the analytic continuation of the classical eigenvalue. We tried to extend their description to the non classical CM eigenvalues. We have mainly some conjectures based on numerical experiments and, even in the classical case, a fascinating conjecture (in Bender-Wu style) remains open. 
  
6. Analytic spectrums. 

I will sketch a “new” spectral theory based on the wild monodromy representations. The basic heuristic is that, up to an equivalence of tannakian categories, a rational linear equation“is” its wild monodromy representation and therefore it must be possible to define its `natural spectrums” using only this representation. I will explicit the things elementary.

In fact the idea is very natural and it appears between the lines in classical studies of some special functions and in some works in physics (as linear perturbation of black-holes) and quantum chemistry. In particular in the fundamental article of Schr\”odinger on the hydrogen atom. 

7. Interpretation of the prolate operators and their spectrums in terms of algebraic geometry. Wild Riemann-Hilbert correspondance. 

The (wild) Riemann-Hilbert correspondance RH associates to an equation its (wild) monodromy representation up to equivalence. An algebraic version of the quotient is an algebraic variety, the (wild) character variety.

In the general prolate case, this wild character variety is a family of affine cubic hypersurfaces. For the $W_\Lambda$ family case, we have such a cubic surface, there are natural coordinates and we observe that this cubic hypersurface $\chi$ is defined on $\mathbb{Z}$. The classical spectrum corresponds to the inverse image by RH of a (double) line on  $\chi$ which is defined on $\mathbb{Z}$. The non-classical CM-spectrum corresponds to the inverse image by RH of an (explicitely calculable) algebraic curve on $\chi$. This algebraic curve is defined on $\mathbb{Z}$.