20 research outputs found
Magnetic spectral bounds on starlike plane domains
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 says that is maximal for a
disk whenever is concave increasing, , the domain has area
, and is the -th Dirichlet eigenvalue of the magnetic
Laplacian . Here the flux
is constant, and the scale invariant factor penalizes deviations
from roundness, meaning for all domains and for disks
Maximizing Neumann fundamental tones of triangles
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 hence among
all triangles of given area. Similar results are proved for the harmonic and
arithmetic means of the first two nonzero eigenvalues
Sums of magnetic eigenvalues are maximal on rotationally symmetric domains
The sum of the first n energy levels of the planar Laplacian with constant
magnetic field of given total flux is shown to be maximal among triangles for
the equilateral triangle, under normalization of the ratio (moment of
inertia)/(area)^3 on the domain. The result holds for both Dirichlet and
Neumann boundary conditions, with an analogue for Robin (or de Gennes) boundary
conditions too. The square similarly maximizes the eigenvalue sum among
parallelograms, and the disk maximizes among ellipses. More generally, a domain
with rotational symmetry will maximize the magnetic eigenvalue sum among all
linear images of that domain. These results are new even for the ground state
energy (n=1).Comment: 19 pages, 1 figur