Dirichlet eigenfunctions on the cube, sharpening the Courant nodal inequality

This paper is devoted to the refine analysis of Courant's theorem for the Dirichlet Laplacian. Many papers (and some of them quite recent) have investigated in which cases this inequality in Courant's theorem is an equality: Pleijel, Helffer--Hoffmann-Ostenhof--Terracini, Helffer--Hoffmann-Ostenhof, B\'erard-Helffer, Helffer--Persson-Sundqvist, L\'ena, Leydold. All these results were devoted to $(2D)$-cases in open sets in $\mathbb R^2$ or in surfaces like $\mathbb S^2$ or $\mathbb T^2$. The aim of the current paper is to look for analogous results for domains in $\mathbb{R}^3$ and, as $\AA.$Pleijel was suggesting in his 1956 founding paper, for the simplest case of the cube. More precisely, we will prove that the only eigenvalues of the Dirichlet Laplacian which are Courant sharp are the two first eigenvalues

Spectral properties of higher order anharmonic oscillators

We discuss spectral properties of the self-adjoint operator $$-d^2/dt^2 + (t^{k+1}/(k+1)-\alpha)^2$$ in $L^2(\mathbb{R})$ for odd integers $k$. We prove that the minimum over $\alpha$ of the ground state energy of this operator is attained at a unique point which tends to zero as $k$ tends to infinity. Moreover, we show that the minimum is non-degenerate. These questions arise naturally in the spectral analysis of Schr\"{o}dinger operators with magnetic field. This extends or clarifies previous results by Pan-Kwek, Helffer-Morame, Aramaki, Helffer-Kordyukov and Helffer.Comment: 15 pages, 2 figure

Decay of superconductivity away from the magnetic zero set

We establish exponential bounds on the Ginzburg-Landau order parameter away from the curve where the applied magnetic field vanishes. In the units used in this paper, the estimates are valid when the parameter measuring the strength of the applied magnetic field is comparable with the Ginzburg-Landau parameter. This completes a previous work by the authors analyzing the case when this strength was much higher. Our results display the distribution of surface and bulk superconductivity and are valid under the assumption that the magnetic field is H\"older continuous

Courant-sharp eigenvalues for the equilateral torus, and for the equilateral triangle

We address the question of determining the eigenvalues $\lambda\_n$ (listed in nondecreasing order, with multiplicities) for which Courant's nodal domain theorem is sharp i.e., for which there exists an associated eigenfunction with $n$ nodal domains (Courant-sharp eigenvalues). Following ideas going back to Pleijel (1956), we prove that the only Courant-sharp eigenvalues of the flat equilateral torus are the first and second, and that the only Courant-sharp Dirichlet eigenvalues of the equilateral triangle are the first, second, and fourth eigenvalues. In the last section we sketch similar results for the right-angled isosceles triangle and for the hemiequilateral triangle.Comment: Slight modifications and some misprints correcte