64 research outputs found
Ergodic theorems for continuous-time quantum walks on crystal lattices and the torus
We give several quantum dynamical analogs of the classical Kronecker-Weyl
theorem, which says that the trajectory of free motion on the torus along
almost every direction tends to equidistribute. As a quantum analog, we study
the quantum walk starting from a localized initial
state . Then the flow will be ergodic if this evolved state becomes
equidistributed as time goes on. We prove that this is indeed the case for
evolutions on the flat torus, provided we start from a point mass, and we prove
discrete analogs of this result for crystal lattices. On some periodic graphs,
the mass spreads out non-uniformly, on others it stays localized. Finally, we
give examples of quantum evolutions on the sphere which do not equidistribute.Comment: 26 pages, 4 figure
Asymptotics of large eigenvalues for a class of band matrices
We investigate the asymptotic behaviour of large eigenvalues for a class of
finite difference self-adjoint operators with compact resolvent in
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