New eigenfunctions for the negative part of the Connes–Moscovici prolate spectrum

In 1880, C. Niven, motivated by a problem of conduction of heat, studied the Helmoltz equation on a spheroid. We consider only the prolate spheroid case. There is a separation of variables of the Laplacian in corresponding spheroidal coordinates which is adapted to the problem. Then we get three ordinary differential equations. One is elementary. In the non spherical case, the two others are two copies of the same equation. This equation is called the prolate spheroidal equation. In 1998 Alain Connes discovered a self-adjoint extension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mrow> <mml:mi>Λ</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">sa</mml:mi> </mml:mrow> </mml:msub> </mml:math> of the prolate operator (of order zero) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>Λ</mml:mi> </mml:msub> </mml:math> . In 2021 Alain Connes and Henri Moscovici discovered that the restriction of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mrow> <mml:mi>Λ</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">sa</mml:mi> </mml:mrow> </mml:msub> </mml:math> to the complement of the finite interval <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mo>-</mml:mo> <mml:mi>Λ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Λ</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> admits (besides a replica of the classical positive spectrum) negative eigenvalues whose ultraviolet behavior reproduces, for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Λ</mml:mi> <mml:mo>=</mml:mo> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> </mml:mrow> </mml:math> , that of the squares of the (shifted) zeros of the Riemann zeta function. In this note we use an approach different from the CM approach. It is based on complex analytic functions in place of Sturm–Liouville theory. In particular we introduce a (new) theory of analytic spectra. This allows in particular for an explicit definition of spectral determinants which are entire functions of order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>≤</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> whose zeros are the eigenvalues. An accurate computation of these eigenvalues follows. We also discovered a new equivalent definition, very simple, of the non classical part of the CM spectrum. The corresponding eigenvalues are the naive eigenvalues of the operator on the imaginary axis (that is the eigenfunctions are bounded on this axis). We prove that they are negative, which was conjectured by Connes and Moscovici, and we obtain a very quick and efficient method of computation of these eigenvalues.