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We use the Euler-Maclaurin formula and the Feynman-Kac formula to extend our previous method of computation of the spectral action based on the Poisson summation formula. We show how to compute directly the spectral action for the general…
We show that, when considering the scaling factor as an affine variable, the coefficients of the asymptotic expansion of the spectral action on a (Euclidean) Robertson-Walker spacetime are periods of mixed Tate motives, involving relative…
In this thesis, we studied certain mathematical issues related to the computation of the Chamseddine--Connes spectral action on some fundamental noncommutative spectral triples, such as the noncommutative torus and the quantum 3-sphere…
We obtain an explicit formula for the full expansion of the spectral action on Robertson-Walker spacetimes, expressed in terms of Bell polynomials, using Brownian bridge integrals and the Feynman-Kac formula. We then apply this result to…
We consider perturbations of Minkowski space as well as more general spacetimes on which the wave operator $\square_g$ is known to be essentially self-adjoint. We define complex powers $(\square_g-i\varepsilon)^{-\alpha}$ by functional…
We formulate a conjecture concerning spectral factorization of a class of trigonometric polynomials of two variables and prove it for special cases. Our method uses relations between the distribution of values of a polynomial of two…
We give the details of the computation of the Chamseddine-Connes action by combination of a Lichnerowicz formula with the heat kernel expension.
The method of covariant symbols of Pletnev and Banin is extended to space-times with topology $\R^n\times S^1\times ... \times S^1$. By means of this tool, we obtain explicit formulas for the diagonal matrix elements and the trace of the…
This is a report on a joint work [12] with D. Essouabri, C. Levy and A. Sitarz. The spectral action on noncommutative torus is obtained, using a Chamseddine--Connes formula via computations of zeta functions. The importance of a Diophantine…
Modulo the moment asymptotic expansion, the Cesaro and parametric behaviours of distributions at infinity are equivalent. On the strength of this result, we construct the asymptotic analysis for spectral densities, arising from elliptic…
We introduce a method that allows the evaluation of general expressions for the spectral functions of the one-dimensional Hubbard model for all values of the on-site electronic repulsion U. The spectral weights are expressed in terms of…
Following the pioneering work of Duistermaat and Gr\"unbaum, we call a family $\{p_n(x)\}_{n=0}^{\infty}$ of polynomials bispectral, if the polynomials are simultaneously eigenfunctions of two commutative algebras of operators: one…
In the context of lattice walk enumeration in cones, we consider the number of walks in the quarter plane with fixed starting and ending points, prescribed step-set and given length. After renormalization, this number may be interpreted as…
The heat kernel expansion is a very convenient tool for studying one-loop divergences, anomalies and various asymptotics of the effective action. The aim of this report is to collect useful information on the heat kernel coefficients…
Asymptotic expansions of Green functions and spectral densities associated with partial differential operators are widely applied in quantum field theory and elsewhere. The mathematical properties of these expansions can be clarified and…
Experimental extraction of $\beta$-shape functions, C(W), is challenging. Comparing different experimental $\beta$-shapes to each other and to those predicted by theory in a consistent manner is difficult. This difficulty is compounded when…
We prove a realization theorem for rational functions of several complex variables which extends the main theorem of M. Bessmertnyi, "On realizations of rational matrix functions of several complex variables," in Vol. 134 of Oper. Theory…
We obtain new partial results supporting the spectral set conjecture in dimension 1.
This paper provides the details of Remark 5.4 in the author's paper "Askey-Wilson polynomials as zonal spherical functions on the SU(2) quantum group", SIAM J. Math. Anal. 24 (1993), 795-813. In formula (5.9) of the 1993 paper a…
We show that a monic polynomial in a discrete variable $n$, with coefficients depending on time variables $t_1, t_2,...$ is a $\tau$-function for the discrete Kadomtsev-Petviashvili hierarchy if and only if the motion of its zeros is…