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Amel Jadlaoui

Publications and source records attributed to Amel Jadlaoui.

5 recordsLinked to original sources
arXiv2026

Hodge Laplacians on Weighted Simplicial Complexes: Forms, Closures, and Bounded Realizations

We establish operator-norm bounds for discrete Hodge Laplacians on weighted flag complexes of a fixed dimension $n$, whose $k$-simplices are the $(k+1)$-cliques of a weighted graph; essential self-adjointness on natural cores follows, with no completeness or curvature assumption. Dual up/down degrees give Schur-type bounds in every degree. At top degree a unitary conjugation turns $Δ_{n}$ into an adjacency operator on the line-complex plus a diagonal potential, to which Schur's test applies directly. At $n=1$ this is the Anderson--Morley edge-degree bound $\|Δ_{1}\|\le\max\{°u+°v\}$, obtained here on infinite and on weighted graphs, whence $\|Δ_{1}\|\le2d$ in the $d$-regular case. The sharpness analysis is carried out at $n=1$, for the edge block; in higher degrees we prove boundedness and essential self-adjointness, not sharpness. We give a spectral criterion for the attainment of $2d$, together with sufficient conditions: it is attained on every finite $d$-regular bipartite graph, and on every infinite one that is amenable. Amenability cannot be dropped: the $d$-regular tree, $d\ge3$, has exact norm $d+2\sqrt{d-1}<2d$. The standard periodic lattices being amenable, we compute their exact norms from the Bloch symbols: $2d$ in the bipartite cases, against $9$ for the triangular and $16$ for the face-centered cubic lattice, on which it is strict. Finally, an ordered coloring which any countable complex admits reformulates the skew model unitarily on unoriented simplices, the orientation signs becoming a function of the colors alone.

math.SP↗
arXiv2026

Geometric Criteria for Essential Self-Adjointness of Discrete Hodge Laplacians on Weighted Simplicial Complexes

We develop a geometric framework for the essential self-adjointness (ESA) of discrete Hodge Laplacians on weighted simplicial complexes of arbitrary dimension. Three notions of $χ$-completeness are introduced --- \emph{global}, \emph{local by level}, and \emph{local by region} --- which are formally distinct as definitions, with the pairwise logical separations remaining partly conjectural. The main results are: \emph{(i)} ESA of the Gauss--Bonnet operator $D=d+δ$ and the Hodge Laplacian $L=D^2$ on $\bigoplus C_c^i(\Vc)$ under operative geometric hypotheses (existence of cut-offs with finite support and bounded weighted degree), which are implied in particular by each of the three completeness notions; \emph{(ii)} ESA of the individual Laplacian block $L_\ell$ under the corresponding local-by-level hypotheses, via a quadratic-form argument; \emph{(iii)} ESA via a Kato--Rellich coupling decomposition under local-by-region hypotheses, applied to discrete half-spaces in the lattice $\mathbb{Z}^d$ ($d\geq 2$) where the coupling has operator norm $\sqrt{2}$ (graph case, i.e.\ $n=2$ in our convention; see Remark~\ref{rem:n-convention-intro}), with a sufficient condition extending the argument to $n\geq 3$; \emph{(iv)} a divergence criterion $\sum_k w_k=\infty$ guaranteeing ESA of $L$ even in the absence of $χ$-completeness, with a worked example (a polynomial-branching tree) where ESA holds yet $χ$-completeness fails. The relationship between the three notions is partly clarified: global $χ$-completeness implies local $χ$-completeness at every level (Remark~\ref{rem:hierarchy-direct}); the converse non-implications are formulated as conjectures, motivated by uniform energy lower bounds in concrete examples (Section~\ref{sec:examples}). The framework recovers and extends classical results for graphs and triangulations, including the optimal divergence rate of \cite{BGJ}.

math.SP↗
arXiv2026

Local Lie Theory in Quasi-Banach Lie Algebras: Convergence of the BCH Series and Geometric Implications

We develop a local Lie theory for Lie algebras equipped with a quasi-norm, i.e., complete topological vector spaces satisfying a relaxed triangle inequality $\|x+y\|\le \Ctri(\|x\|+\|y\|)$ with $\Ctri\ge 1$. We prove that the Baker--Campbell--Hausdorff (BCH) series converges in a neighborhood of the origin, provided the quasi-norm admits a continuous Lie bracket with finite continuity constant $\Cbracket$. The proof relies on the Aoki--Rolewicz theorem to construct an equivalent $p$-norm satisfying $p$-subadditivity, enabling rigorous Cauchy-sequence arguments in the complete quasi-metric space $(E, d_p)$. This yields a well-defined local Lie group structure via the exponential map. We analyze the geometric deformation induced by the quasi-norm exponent $p\in(0,1]$, showing that it modifies metric properties while preserving the underlying Lie algebraic structure. Numerical estimates of BCH coefficients up to degree $20$, with coefficients defined precisely via Hall--Lyndon basis projection, demonstrate that classical combinatorial bounds are conservative in the presence of algebraic cancellations, allowing significantly larger practical convergence radii in structured algebras. Applications include weak Schatten ideals $\mathcal{L}_{p,\infty}(H)$ for $0<p<1$ and certain Hardy-space operator algebras. \smallskip\noindent\textbf{Remark on the convergence radius.} The Catalan-majorant method yields convergence for $\|x\|+\|y\| < 1/(4\Cbracket)$; the additional factor $\Ctri$ appearing in the combined constant $\Ctotal = \Ctri\Cbracket$ is an artefact of switching to the $p$-norm to establish Cauchyness of partial sums. When the quasi-norm itself is directly a $p$-norm ($\Ctri=1$), no such penalty arises and the radius reduces to $1/(4\Cbracket)$.

math.FA↗
arXiv2025

Limiting absorption principle for long-range perturbation in a graphene setting

In this paper, we examine the discrete Laplacian acting on a hexagonal lattice by introducing long-range modifications in both the metric and the potential. Our objective is to establish a Limiting Absorption Principle, excluding possible embedded eigenvalues. To this end, we employ the positive commutator technique as our method.

math-ph↗