arXiv subjects

Matteo Capoferri

Publications and source records attributed to Matteo Capoferri.

At least 19 recordsLinked to original sources

arXiv2026

Higher order Weyl coefficients for the operator curl

We establish refined spectral asymptotics for the operator curl acting on a connected oriented closed Riemannian 3-manifold. We treat positive and negative eigenvalues separately and obtain explicit formulae for the first six global Weyl coefficients. With local Weyl coefficients we compute the first four coefficients as well as the sixth one, and we determine the fifth up to a universal constant. As a consequence, we prove that the eta function of curl (both in its local and global versions) is holomorphic in the complex half-plane $\operatorname{Re}s>-2$. Finally, under appropriate assumptions on the geodesic flow, we improve B\"ar's asymptotic formulae for the positive and negative counting functions, refining the remainder to $o(\lambda^2)$.

math.SP↗

arXiv2025

Anderson localization in high-contrast media with random spherical inclusions

We study spectral properties of partial differential operators modelling composite materials with highly contrasting constituents, comprised of soft spherical inclusions with random radii dispersed in a stiff matrix. Such operators have recently attracted significant interest from the research community, including in the context of stochastic homogenization. In particular, it has been proved that the spectrum of these operators may feature a band-gap structure in the regime where heterogeneities take place on a sufficiently small scale. However, the nature of the limiting (as the small scale tends to zero) spectrum in the above setting is non-classical and not completely understood. In this paper we prove for the first time that Anderson localization occurs near band edges, thus shedding light on the limiting spectral behaviour. Our results rely on recent nontrivial advancements in quantitative unique continuation for PDEs, in combination with assumptions on the model that are standard in the Anderson localization literature, and which we plan to relax in future works.

math.SP↗

arXiv2025

Spectral asymmetry via pseudodifferential projections: the massless Dirac operator

A new approach to the study of spectral asymmetry for systems of partial differential equations (PDEs) on closed manifolds was proposed in a recent series of papers by the first author and collaborator. They showed that information on spectral asymmetry can be encoded within and recovered from a negative order pseudodifferential operator -- the asymmetry operator -- constructed from appropriately defined pseudodifferential (spectral) projections. In this manuscript we apply these techniques to the study of the massless Dirac operator; in particular, we compute the principal symbol of the asymmetry operator, accounting for the underlying gauge invariance.

math-ph↗

arXiv2025

A microlocal pathway to spectral asymmetry: curl and the eta invariant

The notion of eta invariant is traditionally defined by means of analytic continuation. We prove, by examining the particular case of the operator curl, that the eta invariant can equivalently be obtained as the trace of the difference of positive and negative spectral projections, appropriately regularised. Our construction is direct, in the sense that it does not involve analytic continuation, and is based on the use of pseudodifferential techniques. This provides a novel approach to the study of spectral asymmetry of non-semibounded (pseudo)differential systems on manifolds which encompasses and extends previous results.

math.SP↗

arXiv2025

Quantitative estimates for high-contrast random media

This paper studies quantitative homogenization of elliptic equations with random, uniformly elliptic coefficients that vanish in a union of random holes. Assuming an upper bound on the size of the holes and a separation condition between them, we derive optimal bounds for the regularity radius $r_*$ and suboptimal growth estimates for the corrector. These results are key ingredients for error analysis in stochastic homogenization and serve as crucial input for recent developments in the double-porosity model, such as those by Bonhomme, Duerinckx, and Gloria (arXiv:2502.02847).

math.AP↗

arXiv2024

Noncommutative Gelfand Duality: the algebraic case

The goal of this paper is to define a notion of non-commutative Gelfand duality. Using techniques from derived algebraic geometry, we show that the category of rings is anti-equivalent to a subcategory of pre-ringed sites, inspired by Grothendieck's work on commutative rings. Our notion of spectrum, although formally reminiscent of the Grothendieck spectrum, is new. Remarkably, an appropriately refined relative version of our spectrum agrees with the Grothendieck spectrum for finitely generated commutative algebras over the complex numbers, among others. This work aims to represent the starting point for a rigorous study of geometric properties of quantum spacetimes.

math.AG↗

arXiv2024

On boundary conditions for linearised Einstein's equations

We investigate the properties of a fairly large class of boundary conditions for the linearised Einstein equations in the Riemannian setting, ones which generalise the linearised counterpart of boundary conditions proposed by Anderson. Through the prism of the quest to quantise gravitational waves in curved spacetimes, we study their properties from the point of view of ellipticity, gauge invariance, and the existence of a spectral gap.

math.AP↗

arXiv2023

High-contrast random systems of PDEs: homogenisation and spectral theory

We develop a qualitative homogenisation and spectral theory for elliptic systems of partial differential equations in divergence form with highly contrasting (i.e., non uniformly elliptic) random coefficients. The focus of the paper is on the behaviour of the spectrum as the heterogeneity parameter tends to zero; in particular, we show that in general one doesn't have Hausdorff convergence of spectra. The theoretical analysis is complemented by several explicit examples, showcasing the wider range of applications and physical effects of systems with random coefficients, when compared with systems with periodic coefficients or with scalar operators (both random and periodic).

math.AP↗

arXiv2023

Spectral asymptotics for linear elasticity: the case of mixed boundary conditions

We establish two-term spectral asymptotics for the operator of linear elasticity with mixed boundary conditions on a smooth compact Riemannian manifold of arbitrary dimension. We illustrate our results by explicit examples in dimension two and three, thus verifying our general formulae both analytically and numerically.

math.SP↗

arXiv2023

Beyond the Hodge Theorem: curl and asymmetric pseudodifferential projections

We develop a new approach to the study of spectral asymmetry. Working with the operator $\operatorname{curl}:=*\mathrm{d}$ on a connected oriented closed Riemannian 3-manifold, we construct, by means of microlocal analysis, the asymmetry operator -- a scalar pseudodifferential operator of order $-3$. The latter is completely determined by the Riemannian manifold and its orientation, and encodes information about spectral asymmetry. The asymmetry operator generalises and contains the classical eta invariant traditionally associated with the asymmetry of the spectrum, which can be recovered by computing its regularised operator trace. Remarkably, the whole construction is direct and explicit.

math.DG↗

arXiv2022

Two-term spectral asymptotics in linear elasticity

We establish the two-term spectral asymptotics for boundary value problems of linear elasticity on a smooth compact Riemannian manifold of arbitrary dimension. We also present some illustrative examples and give a historical overview of the subject. In particular, we correct erroneous results published in [J. Geom. Anal. 31 (2021), 10164--10193].

math.SP↗

arXiv2022

Topological obstructions to the diagonalisation of pseudodifferential systems

Given a matrix pseudodifferential operator on a smooth manifold, one may be interested in diagonalising it by choosing eigenvectors of its principal symbol in a smooth manner. We show that diagonalisation is not always possible, on the whole cotangent bundle or even in a single fibre. We identify global and local topological obstructions to diagonalisation and examine physically meaningful examples demonstrating that all possible scenarios can occur.

math.AP↗

arXiv2022

Global and microlocal aspects of Dirac operators: propagators and Hadamard states

We propose a geometric approach to construct the Cauchy evolution operator for the Lorentzian Dirac operator on Cauchy-compact globally hyperbolic 4-manifolds. We realise the Cauchy evolution operator as the sum of two invariantly defined oscillatory integrals -- the positive and negative Dirac propagators -- global in space and in time, with distinguished complex-valued geometric phase functions. As applications, we relate the Cauchy evolution operators with the Feynman propagator and construct Cauchy surfaces covariances of quasifree Hadamard states.

math.AP↗

arXiv2021

Diagonalization of elliptic systems via pseudodifferential projections

Consider an elliptic self-adjoint pseudodifferential operator $A$ acting on $m$-columns of half-densities on a closed manifold $M$, whose principal symbol is assumed to have simple eigenvalues. Relying on a basis of pseudodifferential projections commuting with $A$, we construct an almost-unitary pseudodifferential operator that diagonalizes $A$ modulo an infinitely smoothing operator. We provide an invariant algorithm for the computation of its full symbol, as well as an explicit closed formula for its subprincipal symbol. Finally, we give a quantitative description of the relation between the spectrum of $A$ and the spectrum of its approximate diagonalization, and discuss the implications at the level of spectral asymptotics.

math.AP↗

arXiv2021

Eigenfunctions localised on a defect in high-contrast random media

We study the properties of eigenvalues and corresponding eigenfunctions generated by a defect in the gaps of the spectrum of a high-contrast random operator. We consider a family of elliptic operators $\mathcal{A}^\varepsilon$ in divergence form whose coefficients are random, possess double porosity type scaling, and are perturbed on a fixed-size compact domain (a defect). Working in the gaps of the limiting spectrum of the unperturbed operator $\hat{\mathcal{A}}^\varepsilon$, we show that the point spectrum of $\mathcal{A}^\varepsilon$ converges in the sense of Hausdorff to the point spectrum of the limiting two-scale operator $\mathcal{A}^\mathrm{hom}$ as $\varepsilon \to 0$. Furthermore, we prove that the eigenfunctions of $\mathcal{A}^\varepsilon$ decay exponentially at infinity uniformly for sufficiently small $\varepsilon$. This, in turn, yields strong stochastic two-scale convergence of such eigenfunctions to eigenfunctions of $\mathcal{A}^\mathrm{hom}$.

math.SP↗

arXiv2021

Invariant subspaces of elliptic systems I: pseudodifferential projections

Consider an elliptic self-adjoint pseudodifferential operator $A$ acting on $m$-columns of half-densities on a closed manifold $M$, whose principal symbol is assumed to have simple eigenvalues. We show existence and uniqueness of $m$ orthonormal pseudodifferential projections commuting with the operator $A$ and provide an algorithm for the computation of their full symbols, as well as explicit closed formulae for their subprincipal symbols. Pseudodifferential projections yield a decomposition of $L^2(M)$ into invariant subspaces under the action of $A$ modulo $C^\infty(M)$. Furthermore, they allow us to decompose $A$ into $m$ distinct sign definite pseudodifferential operators. Finally, we represent the modulus and the Heaviside function of the operator $A$ in terms of pseudodifferential projections and discuss physically meaningful examples.

math.AP↗

arXiv2021

Invariant subspaces of elliptic systems II: spectral theory

Consider an elliptic self-adjoint pseudodifferential operator $A$ acting on $m$-columns of half-densities on a closed manifold $M$, whose principal symbol is assumed to have simple eigenvalues. We show that the spectrum of $A$ decomposes, up to an error with superpolynomial decay, into $m$ distinct series, each associated with one of the eigenvalues of the principal symbol of $A$. These spectral results are then applied to the study of propagation of singularities in hyperbolic systems. The key technical ingredient is the use of the carefully devised pseudodifferential projections introduced in the first part of this work, which decompose $L^2(M)$ into almost-orthogonal almost-invariant subspaces under the action of both $A$ and the hyperbolic evolution.

math.AP↗

arXiv2020

Global propagator for the massless Dirac operator and spectral asymptotics

We construct the propagator of the massless Dirac operator $W$ on a closed Riemannian 3-manifold as the sum of two invariantly defined oscillatory integrals, global in space and in time, with distinguished complex-valued phase functions. The two oscillatory integrals -- the positive and the negative propagators -- correspond to positive and negative eigenvalues of $W$, respectively. This enables us to provide a global invariant definition of the full symbols of the propagators (scalar matrix-functions on the cotangent bundle), a closed formula for the principal symbols and an algorithm for the explicit calculation of all their homogeneous components. Furthermore, we obtain small time expansions for principal and subprincipal symbols of the propagators in terms of geometric invariants. Lastly, we use our results to compute the third local Weyl coefficients in the asymptotic expansion of the eigenvalue counting functions of $W$.

math.AP↗