Research

2025

1. JPhysA

   Universal Estimates for the Density of States for Aperiodic Block Subwavelength Resonator Systems

   Habib Ammari, Silvio Barandun, Bryn Davies, Erik Orvehed Hiltunen, and Alexander Uhlmann

   Journal of Physics A: Mathematical and Theoretical, Oct 2025

   DOI
2. arXiv

   Uniform Hyperbolicity, Bandgaps and Edge Modes in Aperiodic Systems of Subwavelength Resonators

   Habib Ammari, Clemens Thalhammer, and Alexander Uhlmann

   Sep 2025

   Submitted for publication

   DOI
3. RSPA

   Subwavelength Localization in Disordered Systems

   Habib Ammari, Silvio Barandun, and Alexander Uhlmann

   Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Aug 2025

   Abs DOI

   We elucidate the different mechanisms of wave localization in disordered finite systems of subwavelength resonators, where the disorder is in the spatial arrangement of the resonators. To do so, we employ the capacitance matrix formalism and develop a variety of tools to understand localization in this setting. Namely, we adapt the Thouless criterion of localization to quantify the various mechanisms of localization. We also employ a propagation matrix approach to prove a Saxon–Hutner type theorem ensuring the existence of band gaps for disordered systems and to predict the corresponding Lyapunov exponents. We further investigate the stabilities of bandgaps and localized eigenmodes with respect to fluctuations in the system by introducing a method of defect mode prediction based on the discrete Green function and illuminating the phenomena of Anderson localization and level repulsion under global disorder.

4. arXiv

   Competing Edge and Bulk Localisation in Non-Reciprocal Disordered Systems

   Habib Ammari, Silvio Barandun, Clemens Thalhammer, and Alexander Uhlmann

   Aug 2025

   Submitted for publication

   DOI
5. RSPA

   Generalized Brillouin Zone for Non-Reciprocal Systems

   Habib Ammari, Silvio Barandun, Ping Liu, and Alexander Uhlmann

   Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Mar 2025

   Abs DOI

   Non-reciprocal physical systems, such as waveguides with imaginary gauge potentials or mass-spring chains with active elements, present new mathematical challenges beyond those of ordinary Hermitian structures. In a phenomenon known as the non-Hermitian skin effect, eigenmodes condensate at one edge of the structure and decay exponentially in space. Traditional Floquet–Bloch theory, relying on real-valued quasi-periodicities, fails to capture this decay and thus gives incomplete predictions of spectral behaviour. In this paper, we develop a rigorous theory to address this shortcoming. By extending the Brillouin zone into the complex plane, we introduce the notion of generalized Brillouin zone. We show that this complex formulation reflects the unidirectional decay inherent in non-reciprocal systems and provides an accurate framework for spectral analysis. Our main results are proven in the general context of k-Toeplitz matrices and operators, which model a wide variety of both finite and semi-infinite or infinite non-Hermitian periodic structures. We demonstrate that our generalized Floquet–Bloch formalism correctly identifies spectra and restores spectral convergences for large finite lattices.

6. Sigma

   Tunable Localisation in Parity-Time-Symmetric Resonator Arrays with Imaginary Gauge Potentials

   Habib Ammari, Silvio Barandun, Ping Liu, and Alexander Uhlmann

   Forum of Mathematics, Sigma, Jan 2025

   DOI
7. TLMS

   Truncated Floquet–Bloch Transform for Computing the Spectral Properties of Large Finite Systems of Resonators

   Habib Ammari, Silvio Barandun, and Alexander Uhlmann

   Transactions of the London Mathematical Society, Jan 2025

   Abs DOI

   In subwavelength physics, a challenging problem is to characterise the spectral properties of finite systems of subwavelength resonators. In particular, it is important to identify localised modes as well as bandgaps and associated mobility edges. It was shown numerically that the truncated Floquet–Bloch transform can be used to characterise the spectral properties of finite periodic and some aperiodic large systems of resonators. In this paper, for the first time, the mathematical foundations of this transform are provided. In particular, it is proved that the truncated Floquet–Bloch transform enables an accurate recovery of the structure’s bandgap and defect localised eigenmodes.