Multidimensional Schrödinger operators whose spectrum features a half-line and a Cantor set

We construct multidimensional Schrödinger operators with a spectrum that has no gaps at high energies and that is nowhere dense at low energies. This gives the first example for which this widely expected topological structure of the spectrum in the class of uniformly recurrent Schrödinger operators, namely the coexistence of a half-line and a Cantor-type structure, can be confirmed. Our construction uses Schrödinger operators with separable potentials that decompose into one-dimensional potentials generated by the Fibonacci sequence and relies on the study of such operators via the trace map and the Fricke-Vogt invariant. To show that the spectrum contains a half-line, we prove an abstract Bethe–Sommerfeld criterion for sums of Cantor sets which may be of independent interest

Continuous Spectra For Substitution-Based Sequences

This thesis is chiefly concerned with the continuous spectra of substitution-based sequences. First, motivated by a question of Lafrance, Yee and Rampersad [34], we establish a connection between the ‘root-N’ property and the corresponding sequences that satisfy it having absolutely continuous spectrum. Then we use the recent advances in Bartlett [10, 11] to show that the Rudin–Shapiro-like sequence has singular continuous spectrum, hence does not satisfy the root-N property. This gives a negative answer to the question raised by the authors in [34]. Secondly, we use the connection we establish between the root-N property and absolute continuity to create more substitution-based sequences that have absolutely continuous/Lebesgue spectrum. This is done by modifying Rudin’s original construction [44]. We show that the binary sequences (±1 sequences) from our modification also satisfy the root-N property and they are mutually locally derivable to the corresponding substitution sequences. This shows that the spectral properties of the substitution-based sequences are inherited from their binary counterpart. Finally, we generalise our construction using Fourier matrices. This leads to extending Rudin’s construction to sequences with complex coefficients. This approach allows us to generate substitution sequences of any constant length greater than or equal to two. We show explicitly in the length 3 and 4 cases that these systems exhibit Lebesgue spectrum, employing Bartlett’s algorithm from Chapter 3 and mutual local derivability

Lyapunov Exponents in the Spectral Theory of Primitive Inflation Systems

Manibo CNC. Lyapunov Exponents in the Spectral Theory of Primitive Inflation Systems. Bielefeld: Universität Bielefeld; 2019.In this work, we consider primitive inflation rules as generators of aperiodic tilings, and subsequently, of aperiodic point sets (which are toy models for quasicrystals) deemed adequate for diffraction analysis. We harvest the combinatorial-geometric properties of these systems to obtain renormalisation relations for the pair correlation functions, which carry over to measures that generate the diffraction measure. This yields a measure-valued renormalisation satisfied by each of the components of the diffraction. Using tools from the theory of Lyapunov exponents, we provide a sufficient criterion to rule out the presence of absolutely continuous components in the diffraction and a necessary condition to have a non-trivial absolutely continuous part. Moreover, we provide a computable bound which one can use to use invoke this criterion. We show that this holds for large classes of systems, and, as a sanity check, show that the necessary criterion for existence is satisfied by systems which are a priori known to have absolutely continuous diffraction. Furthermore, we present the recovery of known singularity results and point out connections to number-theoretic quantities which naturally arise from these objects, such as logarithmic Mahler measures