arXiv subjects

M. Aloisio

Publications and source records attributed to M. Aloisio.

5 recordsLinked to original sources

arXiv2026

Eigenfunction correlators under power-law SULE and localization for lattice operators

We develop a deterministic framework showing that a power-law form of semi-uniform localization of eigenfunctions (SULE) imposes strong structural constraints on lattice operators, with consequences of both spectral and dynamical nature. For instance, as spectral consequences we prove that power-law SULE yields geometric constraints on localization centers (such as their equidistribution) and quantitative bounds on eigenfunction correlators. As a dynamical consequence we obtain power-law localization in the sense of finite $q$-moments (up to a certain power $q$) of the position operator. Conversely, suitable bounds on eigenfunction correlators imply a corresponding form of power-law SULE, establishing a close connection between these notions. This highlights the role of power-law SULE as a structural mechanism governing localization beyond the exponential regime, including features typically associated with random operators, such as Anderson-type models. Our results reveal that power-law localization is intrinsically geometric: the spatial distribution of localization centers directly influences eigenfunction correlators and transport properties. As an application, we obtain power-law localization for long-range lattice operators with Stark-type potentials of sublinear growth whose spectral regime exhibits asymptotically collapsing spectral gaps and quasi-resonant structures, without relying on perturbative methods. Applications to long-range random operators are also discussed.

math.SP↗

arXiv2026

Dynamical localization and eigenvalue asymptotics: long-range hopping lattice operators with electric field

We prove power-law dynamical localization for polynomial long-range hopping lattice operators with uniform electric field under any bounded perturbation. Actually, we introduce new arguments in the study of dynamical localization for long-range models with unbounded potentials, involving the Min-Max Principle and a notion of Power-Law ULE. Unlike existing results in the literature, our approach does not rely on KAM techniques or on Green's function estimates, but rather on the asymptotic behavior of the eigenvalues and the potential. It is worth underlining that our general results can be applied to other models, such as Maryland-type potentials.

math.FA↗

arXiv2026

Hölder continuity and Fourier asymptotics of spectral measures for 1D Schrödinger operators under exponentially decaying perturbations

We establish $\frac{1}{2}$-Hölder continuity, or even the Lipschitz property, for the spectral measures of half-line discrete Schrödinger operators under suitable boundary conditions and exponentially decaying small potentials. These are the first known examples, apart from the free case, of Schrödinger operators with Lipschitz continuous spectral measures up to the spectral edge, and it was obtained as a consequence of the Dirichlet boundary condition. Notably, we show that the asymptotic behavior of the time-averaged quantum return probability, either $\log (t) / t$ or $1 / t$, as in the case of the free Laplacian, remains unchanged in this setting. Furthermore, we prove the persistence of the purely absolutely continuous spectrum and the $\frac{1}{2}$-Hölder continuity of the spectral measures for (Diophantine) quasi-periodic operators under exponentially decaying small perturbations. These results are optimal and hold for all energies, up to the border of the absolutely continuous spectrum.

math.SP↗

arXiv2023

On spectral measures and convergence rates in von Neumann's Ergodic Theorem

We show that the power-law decay exponents in von Neumann's Ergodic Theorem (for discrete systems) are the pointwise scaling exponents of a spectral measure at the spectral value~$1$. In this work we also prove that, under an assumption of weak convergence, in the absence of a spectral gap, the convergence rates of the time-average in von Neumann's Ergodic Theorem depend on sequences of time going to infinity.

math.SP↗

arXiv2022

On the Fourier asymptotics of absolutely continuous measures with power-law singularities

We prove sharp estimates on the time-average behavior of the squared absolute value of the Fourier transform of some absolutely continuous measures that may have power-law singularities, in the sense that their Radon-Nikodym derivatives diverge with a power-law order. We also discuss an application to spectral measures of finite-rank perturbations of the discrete Laplacian.

math.SP↗