41 research outputs found
A generalised Gauss circle problem and integrated density of states
Counting lattice points inside a ball of large radius in Euclidean space is a
classical problem in analytic number theory, dating back to Gauss. We propose a
variation on this problem: studying the asymptotics of the measure of an
integer lattice of affine planes inside a ball. The first term is the volume of
the ball; we study the size of the remainder term. While the classical problem
is equivalent to counting eigenvalues of the Laplace operator on the torus, our
variation corresponds to the integrated density of states of the Laplace
operator on the product of a torus with Euclidean space. The asymptotics we
obtain are then used to compute the density of states of the magnetic
Schroedinger operator.Comment: 17 page
Bethe-Sommerfeld Conjecture
We consider Schroedinger operator in () with smooth
periodic potential and prove that there are only finitely many gaps in its
spectrum.Comment: 59 pages, 10 figures; to appear in Annales Henri Poincar
On the principal eigenvalue of a Robin problem with a large parameter
We study the asymptotic behaviour of the principal eigenvalue of a Robin (or
generalised Neumann) problem with a large parameter in the boundary condition
for the Laplacian in a piecewise smooth domain. We show that the leading
asymptotic term depends only on the singularities of the boundary of the
domain, and give either explicit expressions or two-sided estimates for this
term in a variety of situations.Comment: 16 pages; no figures; replaces math.SP/0403179; completely re-writte
Perturbation theory for almost-periodic potentials I. One-dimensional case
We consider the family of operators
in with
almost-periodic potential . We study the behaviour of the integrated density
of states (IDS) when and
is a fixed energy. When is quasi-periodic (i.e. is a finite sum of complex
exponentials), we prove that for each the IDS has a complete
asymptotic expansion in powers of ; these powers are either integer,
or in some special cases half-integer. These results are new even for periodic
. We also prove that when the potential is neither periodic nor
quasi-periodic, there is an exceptional set of energies (which we
call ) such that for any
there is a complete power asymptotic expansion
of IDS, and when , then even two-terms power
asymptotic expansion does not exist. We also show that the super-resonant set
is uncountable, but has measure zero. Finally, we prove that the
length of any spectral gap of has a complete asymptotic
expansion in natural powers of when .Comment: journal version, some misprints are fixed; 28 pages, 1 figur
Fourier transform, null variety, and Laplacian's eigenvalues
We consider a quantity κ(Ω)—the distance to the origin from the null variety of the Fourier transform of the characteristic function of Ω. We conjecture, firstly, that κ(Ω) is maximised, among all convex balanced domains of a fixed volume, by a ball, and also that κ(Ω) is bounded above by the square root of the second Dirichlet eigenvalue of Ω. We prove some weaker versions of these conjectures in dimension two, as well as their validity for domains asymptotically close to a disk, and also discuss further links between κ(Ω) and the eigenvalues of the Laplacians
Classical Wave methods and modern gauge transforms: Spectral Asymptotics in the one dimensional case
In this article, we consider the asymptotic behaviour of the spectral
function of Schr\"odinger operators on the real line. Let have the form
where is a formally self-adjoint first order differential operator with
smooth coefficients, bounded with all derivatives. We show that the kernel of
the spectral projector, , has a complete
asymptotic expansion in powers of . This settles the 1-dimensional case
of a conjecture made by the last two authors.Comment: expanded exposition and simplified various steps proof