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## Multi angle  Regularity of the optimal sets for spectral functionals. Part I: sum of eigenvalues Terracini, Susanna (Auteur de la Conférence) | CIRM (Editeur )

In this talk we deal with the regularity of optimal sets for a shape optimization problem involving a combination
of eigenvalues, under a fixed volume constraints. As a model problem, consider
$\min\Big\{\lambda_1(\Omega)+\dots+\lambda_k(\Omega)\ :\ \Omega\subset\mathbb{R}^d,\ \text{open}\ ,\ |\Omega|=1\Big\},$
where $\langle_i(\cdot)$ denotes the eigenvalues of the Dirichlet Laplacian and $|\cdot|$ the $d$-dimensional Lebesgue measure.
We prove that any minimizer $_{opt}$ has a regular part of the topological boundary which is relatively open and
$C^{\infty}$ and that the singular part has Hausdorff dimension smaller than $d-d^*$, where $d^*\geq 5$ is the minimal
dimension allowing the existence of minimal conic solutions to the blow-up problem.

We mainly use techniques from the theory of free boundary problems, which have to be properly extended to the case of
vector-valued functions: nondegeneracy property, Weiss-like monotonicity formulas with area term; finally through the
properties of non tangentially accessible domains we shall be in a position to exploit the ''viscosity'' approach recently proposed by De Silva.

This is a joint work with Dario Mazzoleni and Bozhidar Velichkov.
In this talk we deal with the regularity of optimal sets for a shape optimization problem involving a combination
of eigenvalues, under a fixed volume constraints. As a model problem, consider
$\min\Big\{\lambda_1(\Omega)+\dots+\lambda_k(\Omega)\ :\ \Omega\subset\mathbb{R}^d,\ \text{open}\ ,\ |\Omega|=1\Big\},$
where $\langle_i(\cdot)$ denotes the eigenvalues of the Dirichlet Laplacian and $|\cdot|$ the $d$-dimensional Lebesgue m...

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## Multi angle  Estimates on the molecular dynamics for the predissociation process Martinez, André (Auteur de la Conférence) | CIRM (Editeur )

We study the survival probability associated with a semiclassical matrix Schrödinger operator that models the predissociation of a general molecule in the Born-Oppenheimer approximation. We show that it is given by its usual time-dependent exponential contribution, up to a reminder term that is small in the semiclassical parameter and for which we find the main contribution. The result applies in any dimension, and in presence of a number of resonances that may tend to infinity as the semiclassical parameter tends to 0.
This is a joint work with Ph. Briet.
We study the survival probability associated with a semiclassical matrix Schrödinger operator that models the predissociation of a general molecule in the Born-Oppenheimer approximation. We show that it is given by its usual time-dependent exponential contribution, up to a reminder term that is small in the semiclassical parameter and for which we find the main contribution. The result applies in any dimension, and in presence of a number of ...

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## Multi angle  Localization of eigenfunctions via an effective potential Jerison, David (Auteur de la Conférence) | CIRM (Editeur )

We discuss joint work with Doug Arnold, Guy David, Marcel Filoche and Svitlana Mayboroda. Consider the Neumann boundary value problem for the operator $L = divA\nabla + V$ on a Lipschitz domain $\Omega$ and, more generally, on manifolds with and without boundary. The eigenfunctions of $L$ are often localized, as a result of disorder of the potential $V$, the matrix of coefficients $A$, irregularities of the boundary, or all of the above. In earlier work, Filoche and Mayboroda introduced the function $u$ solving $Lu = 1$, and showed numerically that it strongly reflects this localization. In this talk, we deepen the connection between the eigenfunctions and this landscape function $u$ by proving that its reciprocal $1/u$ acts as an effective potential. The effective potential governs the exponential decay of the eigenfunctions of the system and delivers information on the distribution of eigenvalues near the bottom of the spectrum. We discuss joint work with Doug Arnold, Guy David, Marcel Filoche and Svitlana Mayboroda. Consider the Neumann boundary value problem for the operator $L = divA\nabla + V$ on a Lipschitz domain $\Omega$ and, more generally, on manifolds with and without boundary. The eigenfunctions of $L$ are often localized, as a result of disorder of the potential $V$, the matrix of coefficients $A$, irregularities of the boundary, or all of the above. In ...

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