arXiv subjects

Raphaël Henry

Publications and source records attributed to Raphaël Henry.

7 recordsLinked to original sources

arXiv2026

The complexity of smooth words over binary alphabets

Smooth words over an alphabet of non-negative integers $\{a,b\}$ are infinite words that are infinitely derivable, the emblematic example being the Oldenburger-Kolakoski word over $\{1,2\}$. The main way to study their language is to consider a finite version of smooth words that we call f-smooth words. In this paper we prove that the f-smooth words are exactly the factors of smooth words, and we make progress towards the conjecture of Sing that the complexity of f-smooth words over $\{a,b\}$ grows like $Θ\left(n^{\log(a+b)/\log((a+b)/2)}\right)$: we prove it over even alphabets, we prove the lower bound over any binary alphabet and we improve the known upper bound over odd alphabets.

cs.FL↗

arXiv2026

On the minimal components of substitution subshifts

In this paper we study substitutions on $A^\mathbb{Z}$ where $A$ is a finite alphabet. We precisely characterize the minimal components of substitution subshifts, give an optimal bound for their number and describe their dynamics. The explicitness of these results provides a method to algorithmically compute and count the minimal components of a given substitution subshift.

math.DS↗

arXiv2016

Spectral analysis of a complex Schrödinger operator in the semiclassical limit

We consider the Dirichlet realization of the operator $-h^2Δ+iV$ in the semi-classical limit $h\to0$, where $V$ is a smooth real potential with no critical points. For a one dimensional setting, we obtain the complete asymptotic expansion, in powers of $h$, of each eigenvalue. In two dimensions we obtain the left margin of the spectrum, under some additional conditions.

math-ph↗

arXiv2014

On the semi-classical analysis of Schrödinger operators with purely imaginary electric potentials in a bounded domain

In this paper, we describe the leftmost eigenvalue of the non-selfadjoint operator $\mathcal{A}_h = -h^2Δ+iV(x)$ with Dirichlet boundary conditions on a smooth bounded domain $Ω\subset\mathbb{R}^n\,$, as $h\rightarrow0\,$. $V$ is assumed to be a Morse function without critical point at the boundary of $Ω\,$. More precisely, we compare $\inf\Reσ(\mathcal{A}_h)$ with the minimum of the spectrum's real part for some model operator. In the case where $V$ has no critical point, the spectrum is determined by the boundary points where $\nabla V$ is orthogonal, and the model operator involves a $1$-dimensional complex Airy operator in $\mathbb{R}^+\,$. If $V$ is a Morse function with critical points in $Ω\,$, the behavior of the operator near the critical points prevails, and the model operator is a complex harmonic oscillator. This question is related to the decay of associated semigroups. In particular, it allows to recover, in a simplified setting, some stability results by Almog in superconductivity theory.

math.SP↗

arXiv2013

Spectral instability for even non-selfadjoint anharmonic oscillators

We study the instability of the spectrum for a class of non-selfadjoint anharmonic oscillators, estimating the behavior of the instability indices (i. e. the norm of spectral projections) associated with the large eigenvalues of these oscillators. We get asymptotic expansions for the instability indices, extending the results of Davies and Davies-Kuijlaars.

math.SP↗

arXiv2013

Spectral projections of the complex cubic oscillator

We prove the spectral instability of the complex cubic oscillator $-\frac{d^2}{dx^2}+ix^3+iαx$ for non-negative values of the parameter $α$, by getting the exponential growth rate of $\|Π_n(α)\|$, where $Π_n(α)$ is the spectral projection associated with the $n$-th eigenvalue of the operator. More precisely, we show that for all non-negative $α$ \[ \lim\limits_{n\to+\infty}\frac{1}{n}\log\|Π_n(α)\| = \fracπ{\sqrt{3}}. \]

math.SP↗

arXiv2013

Spectral instability of some non-selfadjoint anharmonic oscillators

The purpose of this Note is to highlight the spectral instability of some non-selfadjoint differential operators, by studying the growth rate of the norms of the spectral projections $Π_n$ associated with their eigenvalues. More precisely, we are concerned with the complex Airy operator and even anharmonic oscillator. We get asymptotic expansions for the norm of the spectral projections associated with the large eigenvalues, extending the results of Davies and Davies-Kuijlaars.

math.SP↗