Related papers: Sums of magnetic eigenvalues are maximal on rotati…
The sum of the first $n \geq 1$ eigenvalues of the Laplacian is shown to be maximal among triangles for the equilateral triangle, maximal among parallelograms for the square, and maximal among ellipses for the disk, provided the ratio…
The sum of the first $n \geq 1$ eigenvalues of the Laplacian is shown to be maximal among simplexes for the regular simplex (the regular tetrahedron, in three dimensions), maximal among parallelepipeds for the hypercube, and maximal among…
We develop sharp upper bounds for energy levels of the magnetic Laplacian on starlike plane domains, under either Dirichlet or Neumann boundary conditions and assuming a constant magnetic field in the transverse direction. Our main result…
This paper aims to show that, in the limit of strong magnetic fields, the optimal domains for eigenvalues of magnetic Laplacians tend to exhibit symmetry. We establish several asymptotic bounds on magnetic eigenvalues to support this…
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…
We investigate how the lowest eigenvalue of a magnetic Laplacian depends on the geometry of a planar domain with a disk shaped hole, where the magnetic field is generated by a singular flux. Under Dirichlet boundary conditions on the inner…
We consider the magnetic Robin Laplacian with a negative boundary parameter. Among a certain class of domains, we prove that the disk maximizes the ground state energy under the fixed perimeter constraint provided that the magnetic field is…
We prove sharp isoperimetric inequalities for Neumann eigenvalues of the Laplacian on triangular domains. The first nonzero Neumann eigenvalue is shown to be maximal for the equilateral triangle among all triangles of given perimeter, and…
We obtain upper bounds for the first eigenvalue of the magnetic Laplacian associated to a closed potential $1$-form (hence, with zero magnetic field) acting on complex functions of a planar domain $\Omega$, with magnetic Neumann boundary…
The magnetic Laplacian on a planar domain under a strong constant magnetic field has eigenvalues close to the Landau levels. We study the case when the domain is a disc and the spectrum consists of branches of eigenvalues of one dimensional…
We prove various estimates for the first eigenvalue of the magnetic Dirichlet Laplacian on a bounded domain in two dimensions. When the magnetic field is constant, we give lower and upper bounds in terms of geometric quantities of the…
Among all triangles of given diameter, the equilateral triangle is shown to minimize the sum of the first $n$ eigenvalues of the Dirichlet Laplacian, for each $n \geq 1$. In addition, the first, second and third eigenvalues are each proved…
We consider the first eigenvalue of the magnetic Laplacian in a bounded and simply connected planar domain, with uniform magnetic field and Neumann boundary conditions. We investigate the reverse Faber-Krahn inequality conjectured by S.…
We consider a compact Riemannian manifold M endowed with a potential 1-form A and study the magnetic Laplacian associated with those data (with Neumann magnetic boundary condition if the bpoundary of M is not empty). We first establish a…
Motivated by relativistic materials, we develop a numerical scheme to support existing or state new conjectures in the spectral optimisation of eigenvalues of the Dirac operator, subject to infinite-mass boundary conditions. We study the…
We consider the eigenvalues of the magnetic Laplacian on a bounded domain $\Omega$ of $\mathbb R^2$ with uniform magnetic field $\beta>0$ and magnetic Neumann boundary conditions. We find upper and lower bounds for the ground state energy…
It has recently been conjectured by Bogosel, Henrot, and Michetti that the second positive eigenvalue of the Neumann Laplacian is maximized, among all planar convex domains of fixed perimeter, by the rectangle with one edge length equal to…
The third eigenvalue of the Robin Laplacian on a simply-connected planar domain of given area is bounded above by the third eigenvalue of a disjoint union of two disks, provided the Robin parameter lies in a certain range and is scaled in…
Fix two parallel circles in $\mathbb{R}^3$ centered about a common axis. Among surfaces of revolution immersed in $\mathbb{R}^3$ whose boundary is given by these circles, there is one which maximizes the first Dirichlet eigenvalue. If the…
We consider the magnetic Laplacian with the homogeneous magnetic field in two and three dimensions. We prove that the $(k+1)$-th magnetic Neumann eigenvalue of a bounded convex planar domain is not larger than its $k$-th magnetic Dirichlet…