arXiv subjects

Jiayu Qiu

Publications and source records attributed to Jiayu Qiu.

At least 19 recordsLinked to original sources

arXiv2026

Anderson localisation in infinite and finite resonator crystals with long-range interactions

We establish Anderson localisation for random capacitance operators and quantify concentration of individual modes of finite capacitance matrices for clusters of subwavelength high-contrast resonators. The underlying geometry is fully periodic in dimensions $d=2,3$, with one resonator per cell, and the inverse material factors are independent and uniformly distributed. We retain the full capacitance interaction, whose coefficients decay exponentially with lattice separation. Under coordinate reflection symmetry, spectral and dynamical localisation hold near the upper spectral edge for every fixed positive disorder strength. At weak disorder $W$, we obtain an explicit localisation window of energy width proportional to $W$; prescribed positive intervals are localised at sufficiently large disorder. A variational boundary comparison and a material-variable eigenvalue-spacing estimate transfer localisation from truncated matrices to individual modes of the finite capacitance matrix. In a bounded sample of cell size $\varepsilon$, with probability tending to one, all except for $O(\varepsilon^{1-d}\log(1/\varepsilon))$ modes in the localisation interval concentrate within radii $O(\varepsilon\log(1/\varepsilon))$. The estimates are uniform for the contrast-dependent capacitance family at small contrast. We also prove a common limiting density of states for the truncated and finite matrices and illustrate concentration numerically.

math-ph↗

arXiv2026

Waveguiding in systems of high contrast resonators: Theory and fast computations

In this work, we study guided modes in systems of high-contrast resonators near nonzero interior Neumann frequencies, beyond the subwavelength regime. In the regular exterior regime, where the exterior Dirichlet problem is well-posed at the reference wavenumber, we introduce an infinite-dimensional frequency-dependent capacitance operator obtained by compressing the exterior Helmholtz Dirichlet-to-Neumann map to the traces of the interior resonant Neumann eigenspaces. We prove the norm-resolvent convergence of the continuous problem to this discrete effective operator as the contrast $δ\to0$, and derive first-order asymptotic formulas for compact-defect frequencies and line-defect band functions. We then establish exponential off-diagonal decay of the capacitance coefficients by a Combes--Thomas argument, yielding an exponentially accurate truncation of the discrete operator, and show that its retained coefficients can be computed from local Helmholtz problems. At the physical frequency, this local approximation converges exponentially under a uniform stability assumption for the growing finite-cluster problems. The stability assumption can be removed by introducing a vanishing complex absorption together with a Hermitian symmetrization. In particular, an absorption parameter of order $\sqrtδ$, together with interaction truncation and patch radii of order $|\logδ|$, suffices to preserve the $O(δ^2)$ accuracy of the first-order high-contrast expansion of the defect eigenfrequencies, yielding a fast computational method. Numerical experiments for dipole and quadrupole resonances illustrate the accuracy, exponential locality, and applicability of the discrete model to straight and bent waveguides generated by material or geometric detuning.

math.NA↗

arXiv2026

The concentration conjecture for Dirac--Coulomb operators

We prove the concentration conjecture for the three-dimensional Dirac operator with mass $m>0$ and an attractive Coulomb potential generated by finitely many point charges. For a fixed total charge $G\in(0,1]$, the lowest eigenvalue in the spectral gap is at least $m\sqrt{1-G^2}$, with equality precisely when all the charge is concentrated at one point. The critical one-center operator is understood as its distinguished extension. The result also extends to arbitrary nonnegative finite charge measures. The proof uses a strict two-center estimate for the weighted free Dirac resolvent and a Birman-Schwinger comparison. We establish the two-center estimate through an auxiliary axial operator, combining a zero-energy bound and a scalar resolvent estimate with a radial decomposition, an angular Schur complement, and exact polynomial inequalities.

math-ph↗

arXiv2026

Absence of Absolutely Continuous Spectrum for One-Dimensional Capacitance Operators with Critical Long-Range Interactions

In this paper, we establish the absence of absolutely continuous spectrum for scalar one-dimensional Laurent operators with multiplicative disorder and critical long-range hoppings. In particular, the requirement of at least quartic off-diagonal decay rate in the existing theory is bypassed by exploiting the finite-difference bound of the convolution kernel, which is available even when the interaction decays only as $|n|^{-1}\log^{-β}|n|$ with $β>1$. We then apply the abstract main result to a concrete physical model, the capacitance operator in subwavelength physics. For a chain of three-dimensional resonators with one resonator per period, we verify the required decay and finite-difference estimates and conclude the absence of absolutely continuous spectrum. These results leave open the distinction between pure point and singular continuous spectrum, but constitute a key step in proving Anderson localization in wave systems with critical long-range interactions.

math-ph↗

arXiv2026

On the Dirac Faber-Krahn Conjecture

In this paper, we prove the strict Faber-Krahn inequality for non-negative-mass Dirac operators on simply connected domains with infinite-mass boundary condition. Denoting the first positive eigenvalue of the corresponding Dirac operator with mass $m\geq 0$ as $λ_1^{+}(Ω;m)$, our main result shows that \begin{equation*} λ_1^{+}(Ω;m)\geq λ_1^{+}(B_{|Ω|};m), \end{equation*} where $B_{|Ω|}$ is the disk with the same area as the domain $Ω$, and the equality is strictly valid if and only if $Ω=B_{|Ω|}$. The key observation leading to the proof is that the upper component of the Dirac spinor associated with $λ_1^{+}(Ω;m)$ is non-vanishing over $Ω$, which is proved by exploiting its Dirac current and the associated stream function. Based on this global non-vanishing property, we construct a sphere-valued spinor map of degree one, which allows a sharp comparison with the radial disk profile using the planar isoperimetry.

math.SP↗

arXiv2026

Robustness of Valley-Hall Interface Modes Against Sharp Bending

It is well known that band inversion across a straight interface in a periodic medium gives rise to interface modes that are localized near the interface and propagate along it inside the bulk spectral gap. This phenomenon constitutes the key mechanism underlying the valley-Hall effect. In this paper, we address the long-standing problem of the robustness of such interface modes. We prove that, when the interface is bent through an angle of $\frac{2π}{3}$, the interface modes persist for every frequency in the bulk spectral gap where the group velocity is non-vanishing, except for a finite exceptional set. We also show that corner-localized modes, if they occur, can appear only at these exceptional frequencies and have finite multiplicity. To the best of our knowledge, this is the first rigorous mathematical theory of the bending immunity of valley-Hall interface modes.

math-ph↗

arXiv2026

Long-range interactions and Anderson localisation for one-dimensional high-contrast resonator chain

Spectral and transport properties of high-contrast resonator systems can be described in the subwavelength regime in terms of the so-called capacitance operator. In this paper, we consider an infinitely periodic chain of high-contrast resonators in three dimensions. The first result is a precise estimate of the off-diagonal decay rate of the capacitance operator $C$. Importantly, we demonstrate that the decay rate is long-range and critical: as $|n-m|\to\infty$, \begin{equation*} C(n,m)\sim \frac{1}{|n-m|\log^2|n-m|}, \end{equation*} which is $\ell^1$ summable but slower than quadratic. This borderline decay of the off-diagonal entries makes the present proof of Anderson localisation with arbitrary disorder, which is observed numerically in this paper, out of reach; we hope that this physical example of classical wave systems with critical long-range interactions provides new insight in the field of Anderson localisation. As the second main result, based on the off-diagonal decay estimate, we prove a strong convergence of the finite capacitance operator, which corresponds to a truncated chain, to the capacitance operator as the size of the truncated chain grows to infinity. Using this strong convergence, we improve the results of [Ammari et al., SIAM J. Math. Anal., 2023 and Bull. London Math Soc., 2025] by presenting a rigorous estimate of the convergence rate of the spectrum.

math-ph↗

arXiv2026

Bulk-Edge Correspondence for Finite Two-dimensional Ergodic Disordered Systems

In this paper, we rigorously prove the bulk-edge correspondence for finite two-dimensional ergodic disordered systems. Specifically, we focus on the short-range Hamiltonians with ergodic disordered on-site potentials. We first introduce the bulk and edge indices, which are both well-defined within the Aizenman-Molchanov mobility gap. On the one hand, the bulk index is the usual Hall conductance, which is a well-studied quantized topological number. On the other hand, the edge index, which characterizes the averaged angular momentum of edge modes in the mobility gap, is uniquely associated with finite systems. Our main result proves that as the sample size tends to infinity, the edge index converges to the bulk index almost surely. Our findings provide a rigorous foundation for the bulk-edge correspondence principle for finite disordered systems. The existence of the Aizenman-Molchanov mobility gap is proved by the geometric decoupling method, introduced by Aizenman and Molchanov [Comm. Math. Phys., 1993], under a rational assumption on the distribution of the random potential. For completeness, all assumptions are checked on a prototypical model for (quantum) anomalous Hall physics.

math-ph↗

arXiv2026

Resolvent Convergence and Patch Approximation for Subwavelength Guided Modes in Non-Periodic Systems of High-Contrast Resonators

This paper develops, analyzes, and validates a fast algorithm for computing guided modes within bent interfaces and non-periodic defects in high-contrast resonator crystals, where the Floquet--Bloch theory is not applicable. We first establish the resolvent convergence of the governing continuous operator to the discrete capacitance operator. This result rigorously justifies the reduction of the continuous spectral problem to a discrete eigenvalue problem. Then, we develop a truncation scheme of the discrete operator, named the patch approximation, and derive a rigorous error estimate for the patch approximation. Finally, we validate the accuracy and efficiency of our scheme through various examples. Our framework provides a general, computationally efficient, and rigorously justified approach to simulate guided modes in non-periodic systems of high-contrast resonators.

math.SP↗

arXiv2026

Symmetry-protected Interface Modes Bifurcated from Double Dirac Cones

We rigorously prove the existence of interface modes in a sharp interface model, which bifurcate from the double Dirac cone as a consequence of the band inversion induced by super-symmetry breaking. The exact number of interface modes are determined. The proof is based on a discrete version of the layer-potential framework. Moreover, we prove that such interface modes are symmetry-protected against perturbations that respect the reflection symmetry.

math-ph↗

arXiv2026

Non-Hermitian Fabry-Pérot Resonances

We characterise non-Hermitian Fabry-Pérot resonances in high-contrast resonator systems and study the properties of their associated resonant modes from continuous differential models. We consider two non-Hermitian effects: the exceptional point degeneracy and the skin effect induced by imaginary gauge potentials. Using the propagation matrix formalism, we characterise these two non-Hermitian effects beyond the subwavelength regime. This analysis allows us to (i) establish the existence of exceptional points purely from radiation conditions and to (ii) prove that the non-Hermitian skin effect applies uniformly across resonant modes, yielding broadband edge localisation.

math-ph↗

arXiv2025

Mathematical Theory for Photonic Hall Effect in Honeycomb Photonic Crystals

In this work, we develop a mathematical theory for the photonic Hall effect and prove the existence of guided electromagnetic waves at the interface of two honeycomb photonic crystals. The guided wave resembles the edge states in electronic systems: it is induced by the topological Hall effect, and the wave propagates along the interface but not in the bulk media. Starting from a symmetric honeycomb photonic crystal that attains Dirac points at the high-symmetry points of the Brillouin zone, $K$ and $K'$, we introduce two classes of perturbations for the periodic medium. The perturbations lift the Dirac degeneracy, forming a spectral band valley at the points $K$ and $K'$ with well-defined topological phase that depends on the sign of the perturbation parameters. By employing the layer potential techniques and spectral analysis, we investigate the existence of guided wave along an interface when two honeycomb photonic crystals are glued together. In particular, we elucidate the relationship between the existence of the interface mode and the nature of perturbations imposed on the two periodic media separated by the interface.

physics.optics↗

arXiv2025

Topological interface modes in aperiodic subwavelength resonator chains

We consider interface modes in block disordered subwavelength resonator chains in one dimension. Based on the capacitance operator formulation, which provides a first-order approximation of the spectral properties of dimer-type block resonator systems in the subwavelength regime, we show that a two-fold topological characterization of a block disordered resonator chain is available if it is of dominated type. The topological index used for the characterization is a generalization of the Zak phase associated with one-dimensional chiral-symmetric Hamiltonians. As a manifestation of the bulk-edge correspondence principle, we prove that a localized interface mode occurs whenever the system consists of two semi-infinite chains with different topological characters. We also illustrate our results from a dynamic perspective, which provides an explicit geometric picture of the interface modes, and finally present a variety of numerical results to complement the theoretical results.

physics.optics↗

arXiv2025

Analysis of nonlinear resonances in resonator crystals: Tight-binding approximation and existence of subwavelength soliton-like solutions

This work provides a mathematical framework for elucidating physical mechanisms for confining waves at subwavelength scales in periodic systems of nonlinear resonators. A discrete approximation in terms of the linear capacitance operator is provided to characterize the nonlinear subwavelength resonances. Moreover, the existence of subwavelength soliton-like localized waves in periodic systems of nonlinear resonators is proven. As a by-product, a tight-binding approximation of the capacitance operator is shown to be valid for crystals of subwavelength resonators. Both full- and half-space crystals are considered. The framework developed in this work opens the door to the study of topological properties of periodic lattices of subwavelength nonlinear resonators, such as the emergence of nonlinearity-induced topological edge states, and to elucidate the interplay between nonlinearity and disorder.

math-ph↗

arXiv2025

On the bifurcation of a Dirac point in a photonic waveguide without band gap openning

Recent progress in topological insulators and topological phases of matter has motivated new methods for the localization of waves in photonic structures. Especially, it is established that a Dirac point of a periodic structure can bifurcate into in-gap eigenvalues if the periodic structure is perturbed differently on the two sides of an interface and if a common band gap can be opened for the two perturbed periodic structures near the Dirac point. The associated eigenmodes are localized near the interface and decay exponentially away from it. This paper addresses the less-known situation when the perturbation only lifts the degeneracy of the Dirac point without opening a band gap. Using a two-dimensional waveguide model, we constructed a wave mode bifurcated from a Dirac point of a periodic waveguide. We show that when the constructed mode couples to an analytically continued Floquet-Bloch mode near the Dirac energy, its eigenvalue acquires a strictly negative imaginary part, making the mode resonant. On the other hand, when the coupling vanishes, the imaginary part of the eigenvalue turns to zero, and the constructed mode becomes an interface mode that decays exponentially away from the interface. The developed method can be extended to other settings, thus providing a clear answer to the problem concerning the bifurcation of Dirac points.

math-ph↗

arXiv2025

Bulk-edge correspondence in finite photonic structure

In this work, we establish the bulk-edge correspondence principle for finite two-dimensional photonic structures. Specifically, we focus on the divergence-form operator with periodic coefficients and prove the equality between the well-known gap Chern number (the bulk invariant) and an edge index defined via a trace formula for the operator restricted to a finite domain with Dirichlet boundary conditions. We demonstrate that the edge index characterizes the circulation of electromagnetic energy along the system's boundary, and the BEC principle is a consequence of energy conservation. The proof leverages Green function techniques and can be extended to other systems. These results provide a rigorous theoretical foundation for designing robust topological photonic devices with finite geometries, complementing recent advances in discrete models.

math-ph↗

arXiv2025

Generalized bulk-interface correspondence for non-quantized spin transport

This paper establishes a rigorous mathematical framework for a generalized bulk-interface correspondence (BIC) in electronic systems with possibly nonconserved spin charge, where the Hamiltonian and spin operator do not commute. We first introduce the bulk spin conductance as a character of the bulk medium, which is defined as a potential-current correlation function and is not quantized if the spin charge is nonconserved. Then we establish the principle of BIC, which states that the difference of bulk spin conductances across an interface equals the sum of two quantities associated with the spin transport along the interface: the spin-drift conductance, which captures spin transport carried by interface modes, and the spin-torque conductance, which accounts for spin generation near the interface due to the non-conservation of spin. Furthermore, when the spin charge is conserved, our result recovers the existing BIC based on the spin Chern number or Fu-Kane-Mele $\mathbb{Z}_2$ index. Our findings demonstrate that the principle of BIC is not restricted to systems with quantized characters and provides new insights into spin transport phenomena.

math-ph↗

arXiv2024

A Mathematical Theory of Integer Quantum Hall Effect in Photonics

This paper investigates interface modes in a square lattice of photonic crystal composed of gyromagnetic particles with $C_{4v}$ point group symmetry. The study shows that Dirac or linear degenerate points cannot occur at the three high-symmetry points in the Brillouin zone where two Bloch bands touch. Instead, a touch point at the M-point has a quadratic degeneracy in the generic case. It is further proved that when a magnetic field is applied to the two sides of an interface in opposite directions, two interface modes supported along that interface can bifurcate from the quadratic degenerate point. These results provide a mathematical foundation for the first experimental realization of the integer quantum Hall effect in the context of photonics.

math-ph↗