GdT EDP et Théorie Spectrale

This talk is devoted to the study of Schrödinger equations in the presence of resonant interactions that can lead to energy transfer. When the domain is a Diophantine torus we prove that, over very long time scales, the majority of small solutions in high regularity Sobolev spaces do not exchange energy from low to high frequencies. We first provide context on Birkhoff normal form approaches to study of the long-time dynamics of the solutions to Hamiltonian partial differential equations. Then, we introduce the induction on scales normal form, central to our proof. Throughout the iteration, we ensure appropriate non-resonance properties while modulating the frequencies (of the linearized system) with the amplitude of the Fourier coefficients of the initial data. Our main challenge is then to address very small divisor problems and to describe the set of admissible initial data.The results are based on a joint work with Joackim Bernier, and an ongoing joint work with Gigliola Staffilani.

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In this talk I am interested in formulas describing the low-lying eigenvalues of the Witten Laplacian $\Delta_V = -h^2\Delta + | V^{\prime} |^2 - h V^{\prime \prime}$. The case where $V$ is a Morse function has been largely studied and here I try to obtain similar results when $V$ has some degeneracy. In the end of the presentation I will also give an example of new behaviors that were not observed in the Morse case.

Nous considérerons l'interaction entre une molécule diatomique et un pulse laser et verrons comment calculer semi-classiquement la probabilité pour qu'elle change d'état rotationnel. Nous nous concentrerons en particulier sur le calcul de l'indexe de phase, crucial pour une prise en compte précise des interférences quantiques. 

In this talk, I will discuss the self-adjointness of the two-dimensional Dirac operator coupled with a singular combination of electrostatic and Lorentz scalar $\delta$-interaction, supported on a closed Lipschitz curve. The main new ingredients are an explicit use of the Cauchy transform on non-smooth curves and a direct link with the Fredholmness of a singular boundary integral operator. This results in a proof of self-adjointness for a new range of coupling constants, which includes and extends all previous results for this class of problems. The study is particularly precise for the case of curvilinear polygons, as the angles can be taken into account in an explicit way. In particular, if the curve is a curvilinear polygon with obtuse angles, then there is a unique self-adjoint realization with domain contained in $H^{1/2}$ for the full range of non-critical coefficients in the transmission condition. The results are based on a joint work with Badreddine Benhellal and Konstantin Pankrashkin.

In this talk we show that spectral shift function can be expressed via (regularised) determinant of Birman-Schwinger operator in the setting suitable for higher order differential operators. We then use this expression to show that the spectral shift function for massless Dirac operator is continuous everywhere except possibly at zero. Behaviour of the spectral shift function at zero is influenced by the presence of zero eigenvalue and/or resonance of the perturbed Dirac operator.

Characterizing topological electronic states in crystals is a formidable goal in condensed matter physics. Such states offer substantial breakthroughs toward a deeper understanding of matter and potential applications ranging from electronics to quantum computing. Their unique electronic features – robust boundary states and quantized bulk electromagnetic responses – derive from topological properties of the wave functions in reciprocal space as a refinement of Bloch band theory. Direct evidence of the band topology traditionally relies on the macroscopic response of the electrons to external electromagnetic fields in ultra-clean samples.

In this seminar, I will introduce an alternative approach to identify topological systems. I will present two experiments (from a theoretical physicist perspective) that image the local response of the electrons to a boundary in the local density of states (LDOS). The first experiment focuses on graphene — a 2D semimetal — imaged by scanning tunneling microscopy. The second experiment emulates a 1D insulator with dielectric resonators inside a microwave cavity. Remarkably, both systems exhibit wavefront dislocations in the LDOS near the boundary. I will show that the dislocation charge is a real-space measure of the reciprocal-space band topology.

In this talk, we consider a bounded domain in the Euclidean plane and examine the Laplacian eigenvalue problem supplemented with specific boundary conditions. A famous conjecture by Berry proposes that in chaotic systems, eigenfunctions resemble random monochromatic waves; however, this behavior is generally not expected in integrable systems. In this talk, we explore the behavior of high-energy eigenfunctions and their connection to Berry’s random wave model. We do so by studying a related property called Inverse Localization, which describes how eigenfunctions can approximate monochromatic waves in small regions of the domain.

Nous nous intéressons à l'opérateur de Dirac dans le plan, en présence d'un champ magnétique constant perpendiculaire au plan. Après un rappel des propriétés spectrales de cet opérateur et de son lien avec l'opérateur de Schrödinger associé, nous étudierons l'influence de deux types de modifications de cet opérateur de Dirac: introduction de bords non bornés ("durs" par des conditions au bord ou "fictifs" par l'ajout d'un potentiel barrière) et ajout d'une perturbation localisée dans une zone bornée du plan. Le but sera notamment d'identifier les résultats connus, les questions ouvertes et les phénomènes à étudier.

Dans cette présentation, nous abordons la phase de Berry et les quantifications semi-classiques en physique des solides. Nous détaillons les méthodes semi-classiques employées pour résoudre ces problèmes.

In this talk, we define the resonances of Dirac operators as poles of the meromorphic continuation of the truncated resolvent and discuss a result about their localization: a kind of Rellich Theorem. Firstly, we consider the case of the free Dirac operator perturbated by a bounded and compactly supported external field. Secondly, we consider the case of the MIT bag model outside a smooth and bounded obstacle.

In this session, we will introduce some semiclassical analysis tools that connect quantum propagation at high frequencies to the underlying classical dynamical system. In the first half, we will do a brief refresher on notions like Hamiltonian systems, rudiments of pseudo-differential calculus, Wigner distributions and their weak limits (semiclassical measures and their refinement at two or more microlocal scales).

In the second half, we will go into some applications in the study of high frequencies of operators with degeneracies (like subelliptic operators) and, more specifically in the focus of this working group, on some recent results (Vacelet (2024), Bal, Becker, Drouot, Fermanian, Lu, Watson (2022) on the propagation of edge states along a curved interface between two topological insulators, where the solution is governed by a semiclassical Dirac operator with variable mass.

This talk is based on a joint work with Michael Hitrik and Martin Vogel. Let

$$P ( x , h D_{x} ; h ) = p ( x , h D_{x} ) + h p_{1} ( x , h D_{x} ) + h^{2} p_{2} ( x , h D_{x} ) + \cdots $$

be a differential operator in the semi-classical limit $0 $<$ h \ll 1$. If the Poisson bracket

$$i^{- 1} \{ p , \overline{p} \} ( x , \xi ) = i^{-1} ( p_{\xi}^{\prime} \cdot \overline{p}^{\prime}_{x} - p^{\prime}_{x} \cdot \overline{p}^{\prime}_{\xi} ) ( x , \xi )$$

does not vanish identically on the zero set of $p$, then $P$ is non-self-adjoint and we know (cf. Hörmander 1960, ...) that $P$ has singular values that are ${\mathcal O} (h^{N} )$ , $\forall N \geq 0$, and even ${\mathcal O} ( \exp - 1 / ( C h ) )$ for some $C $>$ 0$ in the analytic case. Also, $P$ has approximate null solutions concentrated to points in phase space where $p = 0$, $\{ p , \overline{p} \} / i $>$ 0$ and similarly $P^{*}$ has approximate null solutions concentrated to points in phase space where $p = 0$, $\{ p , \overline{p} \} / i $<$ 0$.

One may ask for the precise exponential decay rate of the small singular values. We obtain such results for the $\overline{\partial}$ operator on certain exponentially weighted $L^{2}$ spaces on a torus or on a cylinder, that can be interpretated in terms of quantum tunneling between the regions in the energy surface $p = 0$ where $\{ p , \overline{p} \} / i $>$ 0$ and $\{ p , \overline{p} \} / i $<$ 0$ respectively.