arXiv subjects

Jean-Francois Burnol

Publications and source records attributed to Jean-Francois Burnol.

23 records · Page 2Linked to original sources

arXiv1999

Scattering on the p-adic field and a trace formula

I apply the set-up of Lax-Phillips Scattering Theory to a non-archimedean local field. It is possible to choose the outgoing space and the incoming space to be Fourier transforms of each other. Key elements of the Lax-Phillips theory are seen to make sense and to have the expected interrelations: the scattering matrix S, the projection K to the interacting space, the contraction semi-group Z and the time delay operator T. The scattering matrix is causal, its analytic continuation has the expected poles and zeros, and its phase derivative is the (non-negative) spectral function of T, which is also the restriction to the diagonal of the kernel of K. The contraction semi-group Z is related to S (and T) through a trace formula. Introducing an odd-even grading on the interacting space allows to express the Weil local explicit formula in terms of a ``supertrace''. I also apply my methods to the evaluation of a trace considered by Connes.

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arXiv1998

Spectral Analysis of the local Commutator Operators

The spectral analysis of the (local) conductor operator H = log(|q|) + log(|p|) was shown in a previous paper to be given by the Explicit Formula. I give here the spectral analysis of the commutator operator K = i[log(|p|),log(|q|)] (which shares with H the property of complete dilation invariance). Its spectral function is found to be the derivative of the one of H. It is then deduced that K anticommutes with the Inversion and is a bounded operator. Higher commutators are related in the same manner to the higher derivatives of the Gamma function, which thus all acquire an operator theoretic and spectral interpretation.

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arXiv1998

The Explicit Formula and a Propagator

I give a new derivation of the Explicit Formula for the general number field K, which treats all primes in exactly the same way, whether they are discrete or archimedean, and also ramified or not. In another token, I advance a probabilistic interpretation of Weil's positivity criterion, as opposed to the usual geometrical analogies or goals. But in the end, I argue that the new formulation of the Explicit Formula signals a specific link with Quantum Fields, as opposed to the Hilbert-Polya operator idea (which leads rather to Quantum Mechanics).

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arXiv1998

The Explicit Formula in simple terms

This is a semi-expository paper on the easier aspects of the Explicit Formula for the Riemann Zeta Function. The topics reviewed here include: Weil's criterion for the Riemann Hypothesis and its probabilistic interpretation, various formulations of the contribution corresponding to the real place, Haran's version of the Explicit Formula, and the author's own derivation which puts all places on the same footing. This derivation, an addendum to Tate's Thesis, is in the spirit of Weil's insights towards an adelic understanding of the Explicit Formula. Whereas the analyst would likely formulate the Explicit Formula as a multiplicative convolution, Haran's theorem shows it is also an additive convolution. An intriguing conductor operator is considered whose spectral analysis is equivalent to the Explicit Formula. At a finite place it has a positive cuspidal spectrum. I also comment on the flexibility still left in the Explicit Formula, which shows that it has a symmetry group containing the non-zero rational numbers.

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arXiv1998

Spectral Analysis of the local Conductor Operator

The conductor operator acts on a function through multiplying it with the logarithm of the norm of the variable both in position and in momentum space and adding the outcomes. It makes sense at each completion of an arbitrary number field and arose in previous papers by the author where it was shown to be intimately connected with the Explicit Formula of Analytic Number Theory. I complete here its spectral analysis: the conclusion is that the local contribution to the Explicit Formula expressed as an integral over the critical line is completely equivalent to, indeed precisely realizes this spectral analysis. In this picture the inversion is related to complex conjugation, the additive Fourier transform is closely related to the Tate-Gel'fand-Graev Gamma function on the critical line, and the conductor operator itself is just the logarithmic derivative of the Gamma function, again on the critical line, acting as a multiplier. From this one gets that the conductor operator is not only dilation and Fourier invariant, which was obvious, but also inversion invariant. In this manner, an operator theoretic and spectral interpretation of each local contribution to the Explicit Formula on the critical line has been obtained.

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