Escuela de Física-Matemática 2018
Morning lectures
We will begin the course by considering the one-dimensional free Schrödinger equation with periodic boundary conditions
\[
\left\{
\begin{aligned}
&\partial_t u(t,x) - i \partial_{xx} u(t,x) = 0 \\
&u(t,0) = u(t,1)\\
&\partial_x u(t,0) = \partial_{x} u(t,1) \\
&u(0,x) = u_0(x)\\
\end{aligned}
\right.
\]
for \(x\in\mathbb R\) and \(x\in [0,1]\).
The solution can be found easily by separation of variables.
this is one or the simplest partial differential equations, yet only about 30 years ago,
Micheal Berry [1]
and collaborators [2]
discovered a remarkably complex behaviour which they named after the Victorian scientist
William Henry Fox Talbot [3].
The "Talbot effect" manifests in the system above through a striking pattern of rough "periodicity" in time for simple initial data \(u_0(x)\).
For example, if \(u_0(x)\) is a step function, then
During the course, we will examine how the classical tools of Fourier analysis and the analysis of PDEs can be used to show striking results abour revivals and fractalisation for a variety of time-evolution boundary problems.
The sessions will be devoted to studying five dispersive equations (of first order in time) that support the revival/fractalisation dichotomy, either partially or coompletely.
For this, we will cover subjects such as generalised Fourier expansions, spectral analysis of differential operators, the notation of fractal dimensions and perturbation methods for PDEs.
(horario)
Lecture notes:
Jumps, cusps and fractals, in time-evolution PDEs (with solutions)
Jumps, cusps and fractals, in time-evolution PDEs (without solutions)
Slides of Lyonell's talk
References and further reading.
[1] M. V. Berry, J. Phys. A: Math. Gen. 29 (1996) 6617.
[2] M. V. Berry and S. Klein, J. Mod. Opt. 43 (1996) 2139-2164.
[3] H. F. Talbot, Phil. Mag. 9 (1836) 401-407.
[4] M. B. Erdoğan and N. Tzirakis, Dispersive Partial Differential Equations: Well-posedness and Applications. Cambridge University Press (2016).
[5] I. Rodnianski, Contemp. Math. 255 (2000) 181-188.
[6] L. Boulton, G. Farmakis and B. Pelloni, Proc. R. Soc. A 477 (2021) 202110241.
[7] V. Chousionis, M. B. Erdoğan and N. Tzirakis Proc. Lond. Math. Soc. 110
(2014) 543-564.
[8] L. Boulton, G. Farmakis and B. Pelloni, arXiv:2308.09961 (2023).
[9] L. Boulton, P. Olver, B. Pelloni and D. Smith, Stud. Appl. Math. 147 (2021) 1209-1239.
Spectral theory is an extremely rich field which has found its application in many areas of physics and mathematics. One of the reason which makes it so attractive on the formal level is that it provides a unifying framework for problems in various branches of mathematics, for example partial differential equations, calculus of variations, geometry, stochastic analysis, etc.
The goal of the lecture is to acquaint the students
with spectral methods in the theory of linear differential operators
coming both from modern as well as classical physics,
with a special emphasis put on geometrically induced
spectral properties.
We give an overview of both classical results
and recent developments in the field,
and we wish to always do it by providing a physical interpretation
of the mathematical theorems.
(horario)
Lecture notes: Geometrical aspects of spectral theory
Afternoon lectures
Translation surfaces generalize flat tori to higher genus. In this work, we investigate the geodesic flow on \(\mathbb Z^d\)-covers of compact translation surfaces, assuming the existence of a renormalizing (pseudo-Anosov) map on the cover that lifts from a corresponding map on the base surface. Under these conditions, the natural invariant measures are Maharam measures, which can be interpreted as Lebesgue measure on a deformed version of the original surface. Following recent developments in the application of transfer operator techniques to parabolic dynamics, we demonstrate how these measures can be obtained through the analysis of a (twisted) transfer operator on appropriately chosen (anisotropic) Banach spaces.
This work in progress is in collaboration with Roberto Castorrini, Davide Ravotti, and Yuriy Tumarkin.En esta charla vamos a estudiar el espectro de operadores en \(L^2(0,1)\) que tienen la forma \begin{equation*} if'(x)+V(x)f(x)+f(2\pi)k(x) \end{equation*} donde \(V\) es un potencial acotado y \(k(x)\in L^2(0,2\pi)\). Tales operadores, por medio de un isomorfismo se pueden ver de la forma \begin{equation*} if'(x)+f(2\pi)K(x) \end{equation*} con condiciones de frontera \(f(0)=f(2\pi)\). Estos operadores resultan ser perturbaciones de rango no acotadas del operador autoadjunto \(if'(x)\). El caso anterior se puede generalizar de la siguiente forma: Sean \(A\) un operador autoadjunto con espectro puramente discreto y \(f\) un funcional lineal densamente definido y no acotado, las perturbaciones de rango uno no acotadas de \(A\) vienen de la forma \begin{equation*} A+f(\cdot)\psi. \end{equation*} Bajo ciertas condiciones sobre \(f\) podremos estudiar los autovalores de tales perturbaciones, centrandonos en aspectos como multiplicidades geométricas, algebraicas y comportamiento asintótico de los autovalores en el plano complejo.
(horario)En esta charla hablaremos del problema peri\'odico asociado al sistema de\\ Zakharov-Rubenchik/Benney-Roskes unidimensional \begin{equation*} \left\{ \begin{aligned} & \partial_t\psi-\sigma_3 \partial_x\psi-i \delta \partial_x^2\psi+i\left\{\sigma_2|\psi|^2+W\left(\rho+D \partial_x\phi\right)\right\} \psi=0, \\[1ex] & \partial_t\rho+\partial_x^2\phi+D\partial_x\left(|\psi|^2\right)=0 \\[1ex] & \partial_t\phi+\frac{1}{M^2} \rho+|\psi|^2=0 \end{aligned} \right. \end{equation*} el cual ha aparecido en el contexto de f\'isica de plasma y ondas de agua. Esencialmente, si el tiempo lo permite, mostraremos que podemos tener soluciones del problema lineal en \(H^q(\mathbb{T})\times H^s(\mathbb{T})\times H^{s+1}(\mathbb{T})\), \(q,s \in \mathbb{R}\), y, siguiendo las ideas de \cite{lannes}, \cite{obrecht} y \cite{saut}, soluciones en \(H^s(\mathbb{T})\times H^s(\mathbb{T})\times H^{s+1}(\mathbb{T})\), \(s > 1/2\), de un modelo no lineal modificado.
References and further reading.
[1] David Lannes.
The Water Waves Problem: Mathematical Analysis and Asymptotics,
volume 188 of Mathematical Surveys and Monographs.
AMS, 2013.
[2] Caroline Obrecht.
Sur l'approximation modulationnelle du problème des ondes de surface: Consistance et existence de solutions pour les systèmes de Benney-Roskes/Davey-Stewartson à dispersion exacte}.
Thèse de doctorat, Universit&ecute; Paris Sud - Paris XI, 2015.
NNT: 2015PA112121.
[3]
Jean-Claude Saut and Gustavo Ponce.
Well-posedness for the Benney-Roskes/Zakharov-Rubenchik system.
Discrete Cont. Dynamical Systems, 13(3):811-825, 2005.
In the theory of Schrödinger operators, it is well known that \(\delta\) an \(\delta'\) interactions supported on discrete sets can be modeled via suitable self-adjoint extensions of second-order differential operators, and the spectra of such operators are well understood. However, a natural question arises: what happens when the interactions are supported on a non-discrete set? In this talk, we will present key results concerning the case of a lower bounded potential on the real line, and how the spectrum of the corresponding operator changes when the interactions are supported on a non-discrete set. Finally, we will provide a characterization of the embedded eigenvalues of the perturbed operator.
(horario)
This study highlights the connection between spectral theory and machine learning, offering insights into both the potential and limitations of using deep learning for classical mathematical problems.
Physics-Informed Neural Networks (PINNs) provide a computational framework in which neural networks are trained to solve differential equations by embedding the equation itself, along with boundary, initial, or other constraints, directly into the loss function.
We present a series of computational experiments exploring how PINNs behave when applied to spectral problems and how spectral aspects can help design neural network architectures.
We will examine how PINNs can approximate eigenvalues and eigenfunctions of linear differential operators and observe that their training process tends to favor low-frequency components that are learned more rapidly - a phenomenon known as spectral bias.
In order to improve numerical accuracy in solving PDEs, we incorporate Random Fourier Feature (RFF) into neural networks architectures.
These experiments show a promising intersection between deep learning and spectral aspects.
In this talk, I will present some new spectral properties to a non-selfadjoint Schrödinger operator of the form \(-\frac{d^2}{dx^2}+V\) on a quantum star graph \(\Gamma\) with a non-selfadjoint Robin condition at the central vertex with a complex parameter \(\alpha\) and Dirichlet boundary conditions on the set of outer vertices.
We consider a potential \(V\in L^p(\Gamma;\mathbb C)\) with \(p>2\) and set \((\mu_n(\alpha))_n\) and \((\lambda_n(\alpha))_n\) be the eigenvalues of this Schrödinger operator and of the corresponding unperturbed Laplacian, respectively, repeated according to their algebraic multiplicities such that \((\mathrm{Re}(\lambda_m(\alpha)))_n\) and \( ( \mathrm{Re}(\mu_n(\alpha)))_n\) are monotonically increasing sequences.
More specifically, I will prove that almost all the eigenvalues \(\mu_n(\alpha)\) are simple if we assume that the edge lengths of \(\Gamma\) are incommensurable over \(\{-1,0,1\}\).
Furthermore, we will show that the asymptotic behavior for the real part of these eigenvalues is quadratic.
Finally, we will present the following bounds for the eigenvalues of the Schrödinger operator using the eigenvalues of the Laplacian with \(\alpha=0\) (this is, Neumann-Kirchhoff conditions at the central vertex): \[\lambda_{m-1}(0)+R_m < \mathrm{Re}(\mu_m(\alpha)) < \lambda_{m+1}(0)-R_m\mbox{ as } m\rightarrow \infty\] with \( 0 < R_m=o(m)\).
This is a joint work with E. Leguizamón.
We will discuss the extension theory of dissipative operators of the form \( S+iV\) where \( S\) is a closed symmetric operator and \( V\) is a bounded nonnegative operator. Additionally, we will explore a technique to determine whether these extensions are completely nonselfadjoint. Specifically, we will examine the Schrödinger operator \( S+iV\) on an interval, where \( iV\) is a dissipative multiplication potential. For this case, we will provide a full description of the dissipative extensions in terms of boundary conditions, and we will study their corresponding reducing selfadjoint subspaces, if they exist.
(horario)In this talk we will present some shape optimization problems, associated to low eigenvalue problems for the laplacian with different boundary conditions. We will also make a comment on one Escobar's result regarding the first non-zero eigenvalue for the Stekloff problem.
(horario)I will present recent results on spectral inclusion for operator pencils and block operator matrices, based on several collaborative projects. The findings provide new insights into the interplay between spectral sets and geometric enclosures in the complex plane. Topics include: