Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1510.07019

Dispersion Estimates for the Discrete Laguerre Operator

Aleksey Kostenko·Gerald Teschl
View original source ↗Read source document ↗Export metadata ↓

Abstract

We derive an explicit expression for the kernel of the evolution group $\exp(-\mathrm{i} t H_0)$ of the discrete Laguerre operator $H_0$ (i.e. the Jacobi operator associated with the Laguerre polynomials) in terms of Jacobi polynomials. Based on this expression we show that the norm of the evolution group acting from $\ell^1$ to $\ell^\infty$ is given by $(1+t^2)^{-1/2}$.

Explore related subjects

math-phmath.APmath.CAmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Aleksey Kostenko, Gerald Teschl. 2016-03-16. Dispersion Estimates for the Discrete Laguerre Operator. https://doi.org/10.1007/s11005-016-0831-0

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Gradual eigenvector ergodization in coupled Ginibre matrices

Non-Hermitian random matrices provide a useful framework for understanding universal characteristics of dissipative quantum chaotic systems with loss or gain. We consider a model of two such systems represented by two independent $N\times N$ complex Ginibre matrices interacting via a deterministic matrix $c{\bf 1}_N$, where $c$ is the complex coupling parameter whose magnitude $|c|$ controls the interaction strength. We characterize quantitatively how the eigenvectors of the whole system, initially localized in one of the individual subsystems for $|c|=0$, eventually spread over the full system with growing interaction strength. The resulting asymptotic formula describing such spread in the limit $N\to \infty$ is very explicit and provides a full picture of the gradual ergodization of eigenvectors as a function of the coupling parameter $|c|$ in the whole transition regime. As a by-product of our method we also compute the mean eigenvalue density for our model at the origin of the spectral bulk $z=0$ in the fully ergodic regime, when the coupling is scaled with the matrix size as $c=\sqrt{N}\tilde{c}$. We find that as $N\to \infty$ the limiting density at the origin vanishes beyond the critical value $|\tilde{c}|=1.$ This is compatible with the expected split of the density support in the complex plane into two disjoint domains.

math-ph↗
arXiv2026

Structure-preserving diffuse-domain accelerated saddle dynamics for wetting transitions

Wetting transitions on textured substrates play a central role in the design of functional surfaces, but resolving their transition mechanisms requires efficient exploration of complex energy landscapes. In particular, saddle points are essential for revealing the connectivity among metastable states and transition pathways. In this work, we develop a diffuse-domain accelerated saddle dynamics (DD-ASD) framework for mass-constrained wetting transitions on complex textured substrates. The diffuse-domain formulation embeds complex solid geometries into a Cartesian grid, avoiding body-fitted mesh construction in repeated saddle-point searches. An orthogonal projection is applied consistently to the phase-field state and unstable-direction dynamics, preserving the prescribed droplet mass at the discrete level. Momentum acceleration is incorporated into the saddle dynamics to improve the efficiency of high-index saddle searches. Under the stated local spectral and exact-eigenspace assumptions, we establish a local convergence rate of $1-\mathcal{O}(1/\sqrtκ)$ on the effective mass-conserving subspace. Numerical experiments demonstrate the effectiveness of the proposed method in resolving wetting solution landscapes, transition pathways, and energy barriers on textured substrates. We further extend the framework to fully three-dimensional wetting-landscape computations.

math-ph↗
arXiv2026

Winding Number Statistics of a Parametric Chiral Symplectic Random Matrix Ensemble

The winding number is a simple topological invariant. In the case of chiral symmetry it characterises gapped phases of Fermions. We study statistical properties of this topological index or invariant in a chiral symplectic setting using Random Matrix Theory. We consider ensembles of Hamilton matrices in the symmetry class CII (quaternionic matrices) according to the classification in the tenfold way. We set up a parametric random matrix model and derive expressions for parametric correlations of the winding number density as well as for the discrete winding number distribution. We found a super-universality in the limit of large matrix dimensions for the one- and two-point correlators for the bulk of parameters, meaning that the results agree up to rescaling with those of the class AIII (complex matrices). In this context we employ a new method of unfolding and discover the Gaussian behaviour of the winding number distribution.

math-ph↗