Critical energies in random palindrome models

Abstract

We investigate the occurrence of critical energies—where the Lyapunov exponent vanishes—in random Schrödinger operators when the potentials have some local order, which we call random palindrome models. We give necessary and sufficient conditions for the presence of such critical energies: the commutativity of finite word elliptic transfer matrices. Finally, we perform some numerical calculations of the Lyapunov exponents showing their behavior near the critical energies and the respective time evolution of an initially localized wave packet, obtaining the exponent ruling the algebraic growth of the second momentum. We also consider special random palindrome models with one-letter bounded gap property; the transport effects of such long range order are showed numerically.

Spectral and localization properties for the one-dimensional Bernoulli discrete Dirac operator

“…With respect to nontrivial quantum transport, probed via dynamical delocalization (unbounded moments of the position operator), it was found in random polymer models [18] and in random palindrome models [7] (both including the important random dimer model), due to existence of critical energies [18]. Recently, for 1D discrete Schrödinger operators, Damanik, Sütő and Tcheremchantsev [9] have developed a general method which allows one to derive quantum dynamical lower bounds from upper bounds on the growth of norms of transfer matrices, and they applied this method to some substitution, Sturmian and prime models, among others.…”

Superdiffusive transport in one-dimensional disordered Dirac model

“…, to investigate the delocalization conditions one needs to find the solution of the equation ( ) = 0 E γ . Analysis of ( ) E γ for the dimer model was carried out in [10,14]. It was shown, that for > 0 m Lyapunov exponent ( ) = 0 E γ  , if:…”

Quantum Hamiltonians with quasi-ballistic dynamics and point spectrum

“…Consider the function 2 which is defined for E ∈ (−∞, ∞) and takes values in (0, ∞]. The first result relates G θ,S (E) with the solutions of the eigenvalue equation (9) H 0 θ,S ψ = Eψ. See Theorem 2.4 of [34] for its proof.…”

“…Although not explicitly stated, it is natural to expect that the more "chaotic" the underlining dynamical system, the more singular the corresponding spectrum; the extreme cases could be represented by periodic potentials on one hand, which impose absolutely continuous spectrum and ballistic dynamics (see Definition 1), and random potentials on the other hand, that lead to point spectrum and absence of transport (bounded moments of the position operator). We mention the papers [9,12,13,14,16,17,22,25,37] for references and additional comments on important recent results on quantum dynamics for Dirac and Schrödinger operators.…”