Milivoje Lukic

Quantum Dynamics of Periodic and Limit-Periodic Jacobi and Block Jacobi Matrices with Applications to Some Quantum Many Body Problems

We investigate quantum dynamics with the underlying Hamiltonian being a Jacobi or a block Jacobi matrix with the diagonal and the off-diagonal terms modulated by a periodic or a limit-periodic sequence. In particular, we investigate the transport exponents. In the periodic case we demonstrate ballistic transport, while in the limit-periodic case we discuss various phenomena, such as quasi-ballistic transport and weak dynamical localization. We also present applications to some quantum many body problems. In particular, we establish for the anisotropic XY chain on

On anomalous Lieb–Robinson bounds for the Fibonacci XY chain

{\mathbb{Z}}

Square-summable variation and absolutely continuous spectrum

Z with periodic parameters an explicit strictly positive lower bound for the Lieb–Robinson velocity.

Derivatives of L p Eigenfunctions of Schrödinger Operators

We announce and sketch the rigorous proof of a new kind of anomalous (or sub-ballistic) Lieb-Robinson (LR) bound for an isotropic XY chain in a quasiperiodic transversal magnetic field. Instead of the usual effective light cone |x|≤v|t|, we obtain |x|≤v|t|α for some 0<α<1. We can characterize the allowed values of α exactly as those exceeding the upper transport exponent αu+ of a one-body Schrödinger operator. To our knowledge, this is the first rigorous derivation of anomalous quantum many-body transport. We also discuss anomalous LR bounds with power-law tails for a random dimer field.

Jacobi and CMV Matrices with Coefficients of Generalized Bounded Variation

We rigorously prove a new kind of anomalous (or sub-ballistic) Lieb-Robinson bound for the isotropic XY chain with Fibonacci external magnetic field at arbitrary coupling. It is anomalous in that the usual exponential decay in

The spectrum of a Schrödinger operator with small quasi-periodic potential is homogeneous

x-vt

Schrödinger operators with slowly decaying Wigner–von Neumann type potentials

is replaced by exponential decay in

The isospectral torus of quasi-periodic Schrödinger operators via periodic approximations

x-vt^\alpha