Browsing by Author "Lukic, Milivoje"
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Item Ergodic Schrödinger operators in the infinite measure setting(EMS Press, 2021) Boshernitzan, Michael; Damanik, David; Fillman, Jake; Lukic, MilivojeWe develop the basic theory of ergodic Schrödinger operators, which is well known for ergodic probability measures, in the case of a base dynamics on an infinite measure space. This includes the almost sure constancy of the spectrum and the spectral type, the definition and discussion of the density of states measure and the Lyapunov exponent, as well as a version of the Pastur–Ishii theorem. We also give some counterexamples that demonstrate that some results do not extend from the finite measure case to the infinite measure case. These examples are based on some constructions in infinite ergodic theory that may be of independent interest.Item New Anomalous Lieb-Robinson Bounds in Quasiperiodic XY Chains(American Physical Society, 2014) Damanik, David; Lemm, Marius; Lukic, Milivoje; Yessen, WilliamWe 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 α+u 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.Item Some results on the spectral theory of one-dimensional operators and associated problems(2022-04-21) Young, Giorgio F; Lukic, MilivojeThis thesis discusses results in the area of spectral theory of Schr\"odinger operators, and their discrete analogs, Jacobi matrices, as well as some closely associated problems. The first result we present relates to the quantum dynamics generated by a particular class of almost periodic Schr\"odinger operators. We show that the dynamics generated by Schr\"odinger operators whose potentials are approximated exponentially quickly by a periodic sequence exhibit a strong form of ballistic transport. The second result exploits the connection between the KdV hierarchy and one-dimensional Schr\"odinger operators to prove a uniqueness result for the KdV hierarchy with reflectionless initial data via inverse spectral theoretic techniques. The third and fourth results concern orthogonal and Chebyshev rational functions with poles on the extended real line. In the process of extending some of the existing theory for polynomials and exploring some of the new phenomena that arise, we present a proof of a conjecture of Barry Simon's. This thesis contains joint work with Benjamin Eichinger and Milivoje Luki\'c.Item Wigner-von Neumann type perturbations of periodic Schrödinger operators(American Mathematical Society, 2015) Lukic, Milivoje; Ong, Darren C.Schrödinger operators on the half line. More precisely, the perturbations we consider satisfy a generalized bounded variation condition at infinity and an LP decay condition. We show that the absolutely continuous spectrum is preserved, and give bounds on the Hausdorff dimension of the singular part of the resulting perturbed measure. Under additional assumptions, we instead show that the singular part embedded in the essential spectrum is contained in an explicit countable set. Finally, we demonstrate that this explicit countable set is optimal. That is, for every point in this set there is an open and dense class of periodic Schrödinger operators for which an appropriate perturbation will result in the spectrum having an embedded eigenvalue at that point.