21 research outputs found
Quantum Unique Ergodicity for maps on the torus
When a map is classically uniquely ergodic, it is expected that its
quantization will posses quantum unique ergodicity. In this paper we give
examples of Quantum Unique Ergodicity for the perturbed Kronecker map, and an
upper bound for the rate of convergence.Comment: 17 pages Added a construction of non diophantine irrationals with
arbitrary slow rate of convergenc
The Galois group of random elements of linear groups
Let F be a finitely generated field of characteristic zero and \Gamma<GL_n(F)
a finitely generated subgroup. For an element g in \Gamma, let Gal(F(g)/ F) be
the Galois group of the splitting field of the characteristic polynomial of g
over F. We show that the structure of Gal(F(g)/ F) has a typical behaviour
depending on F, and on the geometry of the Zariski closure of \Gamma (but not
on \Gamma)
Scarred eigenstates for arithmetic toral point scatterers
We investigate eigenfunctions of the Laplacian perturbed by a delta potential
on the standard tori in dimensions .
Despite quantum ergodicity holding for the set of "new" eigenfunctions we show
that there is scarring in the momentum representation for , as well as
in the position representation for (i.e., the eigenfunctions fail to
equidistribute in phase space along an infinite subsequence of new
eigenvalues.) For , scarred eigenstates are quite rare, but for
scarring in the momentum representation is very common --- with denoting the counting function for the new eigenvalues below
, there are eigenvalues corresponding to momentum
scarred eigenfunctions.Comment: 31 pages, 1 figur
On metric diophantine approximation in matrices and Lie groups
We study the diophantine exponent of analytic submanifolds of the space of m
by n real matrices, answering questions of Beresnevich, Kleinbock and Margulis.
We identify a family of algebraic obstructions to the extremality of such a
submanifold, and give a formula for the exponent when the submanifold is
algebraic and defined over the rationals. We then apply these results to the
determination of the diophantine exponent of rational nilpotent Lie groups.Comment: Theorem 3.4 was modified. To appear in Comptes Rendus Mathematiqu
Superscars for arithmetic point scatters II
We consider momentum push-forwards of measures arising as quantum limits (semiclassical measures) of eigenfunctions of a point scatterer on the standard flat torus
. Given any probability measure arising by placing delta masses, with equal weights, on
-lattice points on circles and projecting to the unit circle, we show that the mass of certain subsequences of eigenfunctions, in momentum space, completely localizes on that measure and are completely delocalized in position (i.e., concentration on Lagrangian states). We also show that the mass, in momentum, can fully localize on more exotic measures, for example, singular continuous ones with support on Cantor sets. Further, we can give examples of quantum limits that are certain convex combinations of such measures, in particular showing that the set of quantum limits is richer than the ones arising only from weak limits of lattice points on circles. The proofs exploit features of the half-dimensional sieve and behavior of multiplicative functions in short intervals, enabling precise control of the location of perturbed eigenvalues