Upper bounds in quantum dynamics

Abstract

We develop a general method to bound the spreading of an entire wavepacket under Schrödinger dynamics from above. This method derives upper bounds on time-averaged moments of the position operator from lower bounds on norms of transfer matrices at complex energies. This general result is applied to the Fibonacci operator. We find that at sufficiently large coupling, all transport exponents take values strictly between zero and one. This is the first rigorous result on anomalous transport. For quasi-periodic potentials associated with trigonometric polynomials, we prove that all lower transport exponents and, under a weak assumption on the frequency, all upper transport exponents vanish for all phases if the Lyapunov exponent is uniformly bounded away from zero. By a well-known result of Herman, this assumption always holds at sufficiently large coupling. For the particular case of the almost Mathieu operator, our result applies for coupling greater than two.

The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian

“…We would like to mention that [13] contains the following upper bound for α + u , which holds for λ ≥ 8,…”

New Anomalous Lieb-Robinson Bounds in QuasiperiodicXYChains

“…We mention that α þ u is just one of several transport exponents commonly associated to anomalous one-body dynamics [15,16], but as it turns out, it is the only one relevant for LR bounds.…”

“…A rough numerical study we conducted suggests that α þ u < 1 also holds for 0 ≪ λ < 8, and we think it would be interesting to pursue the numerical aspects further. Moreover, explicit rigorous upper and lower bounds for α þ u exist [13,15,16]. Asymptotically, they behave like ð2 logð1 þ ϕÞÞ= log λ…”

“…Thanks to extensive study, there exist both rigorous and numerical upper and lower bounds on α þ u [10][11][12][13][14][15][16]. We mention that quasiperiodic sequences serve as models for one-dimensional quasicrystals and their sometimes exotic transport properties.…”

“…We mention that quasiperiodic sequences serve as models for one-dimensional quasicrystals and their sometimes exotic transport properties. Especially, the discrete one-body Schrödinger operator with Fibonacci potential, see (5), has been considered [10,[12][13][14][15][16][17][18][19][20][21][22][23][24]. Quasiperiodic spin chains (in particular with Fibonacci disorder) have also been studied extensively, with a focus on spectral properties and critical phenomena [25][26][27][28][29][30][31][32].…”