Erik Hiltunen

• Hiltunen, Erik & Rueff, Liora (2026). Energy Balance and Optical Theorem for Time-Modulated Subwavelength Resonator Arrays. Studies in applied mathematics (Cambridge). ISSN 0022-2526. 156(1). doi: 10.1111/sapm.70168. Full text in Research Archive
• Hiltunen, Erik Orvehed; Kraisler, Joseph; Schotland, John C. & Weinstein, Michael I. (2026). NONLOCAL PARTIAL DIFFERENTIAL EQUATIONS AND QUANTUM OPTICS: BAND STRUCTURE OF PERIODIC ATOMIC SYSTEMS. SIAM Journal on Mathematical Analysis. ISSN 0036-1410. 58(3), p. 2506–2535. doi: 10.1137/25M1754959. Full text in Research Archive
• Wang, Peng; Hiltunen, Erik & Schotland, John C. (2026). Rotating wave approximation and renormalized perturbation theory. Journal of Physics A: Mathematical and Theoretical. ISSN 1751-8113. 59(14). doi: 10.1088/1751-8121/ae551f. Full text in Research Archive
• Bruijn, Yannick De & Hiltunen, Erik Orvehed (2026). Mathematical Foundation for the Generalised Brillouin zone of m-banded Toeplitz operators. arXiv. doi: 10.48550/ARXIV.2602.09734. Full text in Research Archive
• Bruijn, Yannick De & Hiltunen, Erik Orvehed (2026). Complex Brillouin zone for localised modes in Hermitian and non-Hermitian problems. Communications in nonlinear science & numerical simulation. ISSN 1007-5704. 161, p. 110014–110014. doi: 10.1016/j.cnsns.2026.110014. Full text in Research Archive
• Bruijn, Yannick De; Davies, Bryn; Dupuy, Sacha & Hiltunen, Erik Orvehed (2025). Spectra and pseudospectra of non-Hermitian Toeplitz operators: Eigenvector decay transitions in banded and dense matrices. arXiv. doi: 10.48550/ARXIV.2512.03757. Full text in Research Archive
• Bruijn, Yannick De & Hiltunen, Erik Orvehed (2025). Complex Band Structure and localisation transition for tridiagonal non-Hermitian k-Toeplitz operators with defects. arXiv. doi: 10.48550/ARXIV.2505.23610. Full text in Research Archive
• Ammari, Habib; Barandun, Silvio; Cao, Jinghao; Davies, Bryn; Hiltunen, Erik Orvehed & Liu, Ping (2025). The non-Hermitian skin effect with three-dimensional long-range coupling. Journal of the European Mathematical Society (Print). ISSN 1435-9855. doi: 10.4171/jems/1685. Full text in Research Archive Show summary

  We study the non-Hermitian skin effect in a three-dimensional system of finitely many subwavelength resonators with an imaginary gauge potential. We introduce a discrete approximation of the eigenmodes and eigenfrequencies of the system in terms of the eigenvectors and eigenvalues of the so-called gauge capacitance matrix  \mathcal{C}_{N}^{\gamma} , which is a dense matrix due to long-range interactions in the system. Based on translational invariance of this matrix and the decay of its off-diagonal entries, we prove the condensation of the eigenmodes at one edge of the structure by showing the exponential decay of its pseudo-eigenvectors. In particular, we consider a range k -approximation to keep the long-range interaction to a certain extent, thus obtaining a  k -banded gauge capacitance matrix \mathcal{C}_{N,k}^{\gamma} . Using techniques for Toeplitz matrices and operators, we establish the exponential decay of the pseudo-eigenvectors of \mathcal{C}_{N,k}^{\gamma} and demonstrate that they approximate those of the gauge capacitance matrix \mathcal{C}_{N}^{\gamma} well. Our results are numerically verified. In particular, we show that long-range interactions affect only the first eigenmodes in the system. As a result, a tridiagonal approximation of the gauge capacitance matrix, similar to the nearest-neighbour approximation in quantum mechanics, provides a good approximation for the higher modes.

• Ammari, Habib; Barandun, Silvio; Davies, Bryn; Hiltunen, Erik Orvehed & Uhlmann, Alexander (2025). Universal estimates for the density of states for aperiodic block subwavelength resonator systems. Journal of Physics A: Mathematical and Theoretical. ISSN 1751-8113. 58(41). doi: 10.1088/1751-8121/ae0b11. Full text in Research Archive Show summary

  We consider the spectral properties of aperiodic block subwavelength resonator systems in one dimension, with a primary focus on the density of states (DoS). We prove that for random block configurations, as the number of blocks M → ∞ , the integrated DoS converges to a non-random, continuous function. We show both analytically and numerically that the DoS exhibits a tripartite decomposition: it vanishes identically within bandgaps; it forms smooth, band-like distributions in shared pass bands (a consequence of constructive eigenmode interactions); and, most notably, it exhibits a distinct fractal-like character in hybridisation regions. We demonstrate that this fractal-like behaviour stems from the limited interaction between eigenmodes within these hybridisation regions. Capitalising on this insight, we introduce an efficient meta-atom approach that enables rapid and accurate prediction of the DoS in these hybridisation regions. This approach is shown to extend to systems with quasiperiodic and hyperuniform arrangements of blocks.

• Ammari, Habib; Hiltunen, Erik & Rueff, Liora (2025). Space-time wave localization in systems of subwavelength resonators. Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences. ISSN 1364-5021. 481(2307). doi: 10.1098/rspa.2024.0698. Full text in Research Archive Show summary

  In this article, we study the dynamics of metamaterials composed of high-contrast subwavelength resonators and show the existence of localized modes in such a setting. A crucial assumption in this article is time-modulated material parameters. We prove a so-called capacitance matrix approximation of the wave equation in the form of an ordinary differential equation. These formulas set the ground for the derivation of a first-principle characterization of localized modes in terms of the generalized capacitance matrix. Furthermore, we provide numerical results supporting our analytical results showing for the first time the phenomenon of space–time localized waves in a perturbed time-modulated metamaterial. Such spatio-temporal localization is only possible in the presence of subwavelength resonances in the unperturbed structure. We introduce the time-dependent degree of localization to quantitatively determine the localized modes and provide a variety of numerical experiments to illustrate our formulations and results.

• Alexopoulos, Konstantinos; Davies, Bryn & Hiltunen, Erik Orvehed (2025). TOPOLOGICAL INTERFACE MODES IN SYSTEMS WITH DAMPING. SIAM Journal on Applied Mathematics. ISSN 0036-1399. 85(4), p. 1500–1518. doi: 10.1137/24M1710371. Full text in Research Archive
• Hiltunen, Erik (2025). Perturbation theory for dispersion relations of spacetime-periodic materials. Physical review B (PRB). ISSN 2469-9950. 111(14). doi: 10.1103/PhysRevB.111.L140102. Full text in Research Archive Show summary

  We consider Bloch states of weak spacetime-periodic perturbations of homogeneous materials in one spatial dimension. The interplay of the periodicity of space and time leads to an infinitely degenerate dispersion relation in the free case. We consider a general perturbation term, and, as consequence of the infinite degeneracy, we show that the effective equations are given by the eigenvalue problem of an infinite matrix. Our method can be viewed as a time-modulated generalization of the nearly-free-electron model. Based on this result, we find that the infinite degeneracy may split into a family of nondegenerate bands. Our results are illustrated with numerical calculations, and we observe close agreement between the perturbation theory and the numerically computed full solution.

• Davies, B.; Barandun, S.; Hiltunen, Erik; Craster, R.V. & Ammari, H. (2025). Two-scale effective model for defect-induced localization transitions in non-Hermitian systems. Physical review B (PRB). ISSN 2469-9950. 111(3). doi: 10.1103/PhysRevB.111.035109. Full text in Research Archive Show summary

  We illuminate the fundamental mechanism responsible for the transition between the non-Hermitian skin effect and defect-induced localization in the bulk. We study a Hamiltonian with nonreciprocal couplings that exhibits the skin effect (the localization of all eigenvectors at one edge) and add an on-site defect in the center. Using a two-scale asymptotic method, we characterize the long-scale growth and decay of the eigenvectors and derive a simple and intuitive effective model for the transition that occurs when the defect is sufficiently large that one of the modes is localized at the defect site, rather than at the edge of the system.

• Bruijn, Yannick De & Hiltunen, Erik Orvehed (2025). Complex Band Structure for Subwavelength Evanescent Waves. Studies in applied mathematics (Cambridge). ISSN 0022-2526. 154(2). doi: 10.1111/sapm.70022. Full text in Research Archive Show summary

  We present the mathematical and numerical theory for evanescent waves in subwavelength bandgap materials. We begin in the one-dimensional case, whereby fully explicit formulas for the complex band structure, in terms of the capacitance matrix, are available. As an example, we show that the gap functions can be used to accurately predict the decay rate of the interface mode of a photonic analogue of the Su–Schrieffer–Heeger model. In two dimensions, we derive the bandgap Green's function and characterize the subwavelength gap functions via layer potential techniques. By generalizing existing lattice-summation techniques, we illustrate our results numerically by computing the complex band structure in a variety of settings.

• Ammari, Habib; Barandun, Silvio; Davies, Bryn; Hiltunen, Erik & Liu, Ping (2024). STABILITY OF THE NON-HERMITIAN SKIN EFFECT IN ONE DIMENSION. SIAM Journal on Applied Mathematics. ISSN 0036-1399. 84(4), p. 1697–1717. doi: 10.1137/23M1610537. Full text in Research Archive Show summary

  This paper shows both analytically and numerically that the skin effect in systems of non-Hermitian subwavelength resonators is robust with respect to random imperfections in the system. The subwavelength resonators are highly contrasting material inclusions that resonate in a low-frequency regime. The non-Hermiticity is due to the introduction of a directional damping term (motivated by an imaginary gauge potential), which leads to a skin effect that is manifested by the system’s eigenmodes accumulating at one edge of the structure. We elucidate the topological protection of the associated (real) eigenfrequencies and illustrate numerically the competition between the two different localization effects present when the system is randomly perturbed: the non-Hermitian skin effect and the disorder-induced Anderson localization. We show numerically that, as the strength of the disorder increases, more and more eigenmodes become localized in the bulk. Our results are based on an asymptotic matrix model for subwavelength physics and can be generalized also to tight-binding models in condensed matter theory.

• Ammari, Habib; Barandun, Silvio; Davies, Bryn; Hiltunen, Erik; Kosche, Thea & Liu, Ping (2024). Exponentially localized interface eigenmodes in finite chains of resonators. Studies in applied mathematics (Cambridge). ISSN 0022-2526. 153(4). doi: 10.1111/sapm.12765. Full text in Research Archive Show summary

  This paper studies wave localization in chains of finitely many resonators. There is an extensive theory predicting the existence of localized modes induced by defects in infinitely periodic systems. This work extends these principles to finite-sized systems. We consider one-dimensional, finite systems of subwavelength resonators arranged in dimers that have a geometric defect in the structure. This is a classical wave analog of the Su–Schrieffer–Heeger model. We prove the existence of a spectral gap for defectless finite dimer structures and find a direct relationship between eigenvalues being within the spectral gap and the localization of their associated eigenmode. Then, for sufficiently large-size systems, we show the existence and uniqueness of an eigenvalue in the gap in the defect structure, proving the existence of a unique localized interface mode. To the best of our knowledge, our method, based on Chebyshev polynomials, is the first to characterize quantitatively the localized interface modes in systems of finitely many resonators.

• Ammari, Habib; Davies, Bryn & Hiltunen, Erik (2024). Spectral convergence in large finite resonator arrays: The essential spectrum and band structure. Bulletin of the London Mathematical Society. ISSN 0024-6093. doi: 10.1112/blms.13225. Full text in Research Archive Show summary

  We show that the resonant frequencies of a system of coupled resonators in a truncated periodic lattice converge to the essential spectrum of the corresponding infinite lattice. We use the capacitance matrix as a model for fully coupled subwavelength resonators with long-range interactions in three spatial dimensions. For one-, two- or three-dimensional lattices embedded in three-dimensional space, we show that the (discrete) density of states (DOS) for the finite system converges in distribution to the (continuous) DOS of the infinite system. We achieve this by proving a weak convergence of the finite capacitance matrix to corresponding (translationally invariant) Toeplitz matrix of the infinite structure. With this characterisation at hand, we use the truncated Floquet transform to introduce a notion of spectral band structure for finite materials. This principle is also applicable to structures that are not translationally invariant and have interfaces. We demonstrate this by considering examples of perturbed systems with defect modes, such as an analogue of the well-known interface Su–Schrieffer–Heeger model.

• Hiltunen, Erik & Davies, Bryn (2024). Coupled harmonics due to time-modulated point scatterers. Physical review B (PRB). ISSN 2469-9950. 110(18). doi: 10.1103/PhysRevB.110.184102. Full text in Research Archive Show summary

  We consider the resonance and scattering properties of a composite medium containing scatterers whose properties are modulated in time. When excited with an incident wave of a single frequency, the scattered field consists of a family of coupled harmonics at frequencies differing by the frequency of temporal modulation. Similarly, the temporal modulation induces coupling between the resonance frequencies, leading to exceptional points at certain modulation amplitudes. Moreover, the lack of energy conservation causes scattering coefficients to blow up when (complex) resonances cross the real axis. We have developed an integral operator approach to characterize the scattering problem and, for high-contrast scatterers, we present small-volume asymptotic formulas analogous to the classical results for the static (unmodulated) case. We conclude the paper with a boundary integral formulation of the time-modulated problem, which gives an efficient numerical approach and corroborates the asymptotic formulas.

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