37 research outputs found
On the volume of nodal sets for eigenfunctions of the Laplacian on the torus
We study the volume of nodal sets for eigenfunctions of the Laplacian on the
standard torus in two or more dimensions. We consider a sequence of eigenvalues
4\pi^2\eigenvalue with growing multiplicity \Ndim\to\infty, and compute the
expectation and variance of the volume of the nodal set with respect to a
Gaussian probability measure on the eigenspaces. We show that the expected
volume of the nodal set is const \sqrt{\eigenvalue}. Our main result is that
the variance of the volume normalized by \sqrt{\eigenvalue} is bounded by
O(1/\sqrt{\Ndim}), so that the normalized volume has vanishing fluctuations
as we increase the dimension of the eigenspace.Comment: 20 pages, Was accepted for publication in the Annales Henri Poincar
Quantum unique ergodicity for parabolic maps
We study the ergodic properties of quantized ergodic maps of the torus. It is
known that these satisfy quantum ergodicity: For almost all eigenstates, the
expectation values of quantum observables converge to the classical phase-space
average with respect to Liouville measure of the corresponding classical
observable. The possible existence of any exceptional subsequences of
eigenstates is an important issue, which until now was unresolved in any
example. The absence of exceptional subsequences is referred to as quantum
unique ergodicity (QUE). We present the first examples of maps which satisfy
QUE: Irrational skew translations of the two-torus, the parabolic analogues of
Arnold's cat maps. These maps are classically uniquely ergodic and not mixing.
A crucial step is to find a quantization recipe which respects the
quantum-classical correspondence principle. In addition to proving QUE for
these maps, we also give results on the rate of convergence to the phase-space
average. We give upper bounds which we show are optimal. We construct special
examples of these maps for which the rate of convergence is arbitrarily slow.Comment: Latex 2e, revised versio
Value distribution for eigenfunctions of desymmetrized quantum maps
We study the value distribution and extreme values of eigenfunctions for the
``quantized cat map''. This is the quantization of a hyperbolic linear map of
the torus. In a previous paper it was observed that there are quantum
symmetries of the quantum map - a commutative group of unitary operators which
commute with the map, which we called ``Hecke operators''. The eigenspaces of
the quantum map thus admit an orthonormal basis consisting of eigenfunctions of
all the Hecke operators, which we call ``Hecke eigenfunctions''.
In this note we investigate suprema and value distribution of the Hecke
eigenfunctions. For prime values of the inverse Planck constant N for which the
map is diagonalizable modulo N (the ``split primes'' for the map), we show that
the Hecke eigenfunctions are uniformly bounded and their absolute values
(amplitudes) are either constant or have a semi-circle value distribution as N
tends to infinity. Moreover in the latter case different eigenfunctions become
statistically independent. We obtain these results via the Riemann hypothesis
for curves over a finite field (Weil's theorem) and recent results of N. Katz
on exponential sums. For general N we obtain a nontrivial bound on the supremum
norm of these Hecke eigenfunctions