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Soeren Fournais

Publications and source records attributed to Soeren Fournais.

7 recordsLinked to original sources
arXiv2024

Counting eigenvalues below the lowest Landau level

For the magnetic Laplacian on a bounded planar domain, imposing Neumann boundary conditions produces eigenvalues below the lowest Landau level. If the domain has two boundary components and one imposes a Neumann condition on one component and a Dirichlet condition on the other, one gets fewer such eigenvalues than when imposing Neumann boundary conditions on the two components. We quantify this observation for two models: the strip and the annulus. In both models one can separate variables and deal with a family of fiber operators, thereby reducing the problem to counting band functions, the eigenvalues of the fiber operators.

math.SP↗
arXiv2022

Effective operators on an attractive magnetic edge

The semiclassical Laplacian with discontinuous magnetic field is considered in two dimensions. The magnetic field is sign changing with exactly two distinct values and is discontinuous along a smooth closed curve, thereby producing an attractive magnetic edge. Various accurate spectral asymptotics are established by means of a dimensional reduction involving a microlocal phase space localization allowing to deal with the discontinuity of the field.

math.SP↗
arXiv2021

The energy of dilute Bose gases II: The general case

For a dilute system of non-relativistic bosons interacting through a positive potential $v$ with scattering length $a$ we prove that the ground state energy density satisfies the bound $e(ρ) \geq 4πa ρ^2 (1+ \frac{128}{15\sqrtπ} \sqrt{ρa^3} +o(\sqrt{ρa^3}\,))$, thereby proving a lower bound consistent with the Lee-Huang-Yang formula for the energy density. The proof allows for potentials with large $L^1$-norm, in particular, the case of hard core interactions is included. Thereby, we solve a problem in mathematical physics that had been a major challenge since the 1960's.

math-ph↗
arXiv2018

Inequalities for the lowest magnetic Neumann eigenvalue

We study the ground state energy of the Neumann magnetic Laplacian on planar domains. For a constant magnetic field we consider the question whether, under an assumption of fixed area, the disc maximizes this eigenvalue. More generally, we discuss old and new bounds obtained on this problem.

math.SP↗
arXiv2016

Semiclassical Sobolev constants for the electro-magnetic Robin Laplacian

This paper is devoted to the asymptotic analysis of the optimal Sobolev constants in the semiclassical limit and in any dimension. We combine semiclassical arguments and concentration-compactness estimates to tackle the case when an electromagnetic field is added as well as a smooth boundary carrying a Robin condition. As a byproduct of the semiclassical strategy, we also get exponentially weighted localization estimates of the minimizers.

math.AP↗
arXiv2014

Optimal magnetic Sobolev constants in the semiclassical limit

This paper is devoted to the semiclassical analysis of the best constants in the magnetic Sobolev embeddings in the case of a bounded domain of the plane carrying Dirichlet conditions. We provide quantitative estimates of these constants (with an explicit dependence on the semiclassical parameter) and analyze the exponential localization in $L^\infty$-norm of the corresponding minimizers near the magnetic wells.

math.AP↗