We continue our study, initiated in [E. O. Hiltunen et al., SIAM J. Math. Anal., 56 (2024), pp. 3802--3831], of the quantum optics of a single photon interacting with a system of two level atoms. In this work, we investigate the case of a periodic arrangement of atoms. We provide a general structure theorem characterizing the band functions of this problem, which comprise the spectrum of the associated Hamiltonian. Additionally, we study atomic densities arising as periodically arranged scaled inclusions. For this family of examples, we obtain explicit asymptotic formulas for the band functions.

We establish dispersive time-decay estimates for periodic Jacobi operators on the discrete half-line, N. Specifically, we prove t-1/2 decay in the weighted & ell;infinity-1 norm for all such operators. For the global & ell;1 -> & ell;infinity decay estimate, we show that t-1/3 decay holds under a nondegeneracy condition on the discriminant. Alternatively, for any even period q >= 2, if the continuous spectrum consists of exactly q disjoint intervals (bands), we obtain a t-1/(q +1) decay rate without any further assumptions. (c) 2025 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

We study the Schrödinger flow for the SSH model, a class of self-adjoint discrete dimer lattice Hamiltonians on the half-line. Using oscillatory integral techniques, we prove dispersive time-decay estimates, which quantify the spreading of energy throughout the lattice for a localized initial condition. Furthermore, we determine precise dependence of the constants in the decay rates on the parameters of the Hamiltonian. The analysis is complicated by the fact that as a consequence of the boundary condition, the expression for the propagator contains oscillatory integrals with nonintegrable singularities.

We study wave propagation in 2D honeycomb structures with a non-commensurate or “irrational” line defect or edge. Our model is a Schrödinger operator which interpolates, across the edge, between two distinct bulk (asymptotic) Hamiltonians with a common spectral gap about the “Dirac point” of an unperturbed honeycomb operator. We seek edge states, eigenstates that are bounded and oscillatory parallel to the edge, and decaying in the transverse direction. For non-commensurate edges, the rigorous definition of these states is nontrivial due to the lack of translation invariance along the edge. To address this, we exploit quasiperiodicity along the edge by expressing the Hamiltonian as the restriction of a 3D (degenerate elliptic) Hamiltonian describing a 3D medium with a 2D interface within which there is periodicity. Via multiscale analysis, we construct approximate edge states in this 3D setting and obtain by restriction 2D edge states which are quasiperiodic along the irrational edge. These edge states are seeded by eigenfunctions of an effective Dirac operator, which has an infinite block-diagonal structure due to the non-commensurate geometry. A consequence is that infinitely many edge state eigenpairs arise, whose energies are dense in the perturbed bulk spectral gap. In a forthcoming paper, we rigorously construct these gap-filling edge states under a Diophantine condition. The main result here is a key tool in this construction: a resolvent expansion for the 3D Hamiltonian, whose leading term is the resolvent of the block-diagonal Dirac operator. The validity of this expansion requires an omnidirectional non-resonance (no-fold) condition on the dispersion functions of the unperturbed honeycomb Hamiltonian. This condition is satisfied in the strong binding regime. In contrast with earlier works on commensurate edges, the omnidirectional condition is independent of the edge.

This note presents two nontrivial, rotational equilibrium solutions to the spatial uniform gas pressure (isobaric) approximate model of Prosperetti in the inviscid case. Building on Gavrilov's work [GAFA 2019], we first establish the existence of equilibrium solutions with nontrivial (rotational) liquid flow. Second, we construct a nonspherically symmetric, horn-torus-shaped equilibrium bubble under mild spatial decay conditions of the liquid flow. In addition, we extend earlier results on the characterization of spherical equilibrium bubbles to the axisymmetric, purely azimuthal setting. Finally, we implement a numerical simulation of the equilibrium bubble shape using the physics-informed neural network approximation.

The spectral analysis of the unitary monodromy operator, associated with a time-periodically (paramatrically) forced Schrodinger equation, is a question of longstanding interest. Here, we consider this question for Hamiltonians of the form $$H^{\varepsilon}(t)=H^0 + \varepsilon^a W(\varepsilon^a t, -i\nabla)\, ,$$ where $H^0$ is an unperturbed autonomous Hamiltonian, $a\geq 1$, and $W(T,\cdot)$ has a period of $T_{\rm per} >0$. In particular, in the small $\varepsilon>0$ regime, we seek a comparison between the spectral properties of the monodromy operator, the one-period flow map associated with the $H^\varepsilon(t)$ dynamics, and that of the autonomous (unforced) flow, $\exp[-iH^0 T_{\rm per} \varepsilon ^{-a}]$. We consider $H^0$ which is spatially periodic on $\mathbb{R} ^n$ with respect to a lattice. Using the decomposition of $H^0$ and $H^\varepsilon(t)$ into their actions on spaces (Floquet-Bloch fibers) of pseudo-periodic functions, we establish a near spectral-invariance property for the monodromy operator, when acting data which are $\varepsilon$-localized in energy and quasi-momentum. Our analysis requires the following steps: (i) spectrally-localized data are approximated by {\it band-limited (Floquet-Bloch) wavepackets}; (ii) the envelope dynamics of such wavepackets is well approximated by an effective (homogenized) PDE, and (iii) an exact invariance property for band-limited Floquet-Bloch wavepackets, which follows from the effective dynamics. We apply our general results to a number of periodic Hamiltonians, $H^0$, of interest in the study of photonic and quantum materials.

Slowly varying nonuniform strains of non-magnetic wave propagating media with honeycomb symmetry induce an effective- or pseudo-magnetic field, a phenomenon observed first in graphene, and later in photonic crystals and other physical settings. Starting with a discrete nearest-neighbor tight-binding model of a non-uniformly strained honeycomb medium, we derive the continuum effective magnetic Dirac Hamiltonian governing the envelope dynamics of wave packets, which are spectrally localized near a Dirac point (conical band degeneracy) of the unperturbed honeycomb. For unidirectional deformations of bounded gradient, which preserve translation invariance along the ”armchair” direction, we prove the existence of time-harmonic states which are plane-wave like (pseudo-periodic) along the armchair direction and exponentially localized transverse to it. We also obtain the leading order multi-scale structure of such modes for small deformation gradients. Their transverse localization are determined by the eigenstates of a one dimensional effective Dirac Hamiltonian. Our rigorous results apply to deformations which induce an approximate perpendicular constant pseudo-magnetic field (Landau gauge), and yield states with nearly flat band (Landau level) spectrum and hence very high density of states. In contrast, the analogous deformation which preserves translations in the zigzag direction induces no such localization. Corroborating numerical simulations for the different deformation types are presented.

We correct the statement of part (2) of Proposition 4.3, correct typographical errors, and clarify a point concerning equation (B.6) for ρ _g . Moreover, we provide an improved argument for the proof of the main result: Theorem 6.7. The idea of the proof is to reduce the quasilinear problem to a semilinear problem via the implicit function theorem.

We establish a sharp criterion for the stability of a class of compactly supported, homogeneous, symmetric states, "minimal compact solitons" or MCS states, of the time-dependent discrete nonlinear Schrödinger equation on a multilattice, L (L-DNLS). MCS states arise for multilattices where a nearest neighbor, a Laplace-type operator on L, has a flat band. Our stability criterion is in terms of the explicit form of the nonlinearity and the projection of a distinguished vector onto the flat band eigenspace. We apply our general results to MCS states of DNLS with a power-law nonlinearity for the diamond, Kagome, and checkerboard lattices. In lattices where MCS states are unstable, we demonstrate how the variation of the nonlinearity exponent enables the stabilization of small amplitude MCS states.

We present results on quantum tunneling between deep potential wells, in the presence of a strong constant magnetic field. We construct a family of double-well potentials containing examples for which the low-energy eigenvalue splitting vanishes, and hence quantum tunneling is eliminated. Further, by deforming within this family, the magnetic ground state can be made to transition from symmetric to antisymmetric. However, for typical double wells in a certain regime, tunneling is not suppressed, and we provide a lower bound for the eigenvalue splitting.

Edge states are time-harmonic solutions of conservative wave systems which are plane wave-like parallel to and localized transverse to an interface between two bulk media. We study a class of 2D edge Hamiltonians modeling a medium which slowly interpolates between periodic bulk media via a domain wall across a "rational" line defect. We consider the cases of (1) periodic bulk media having the symmetries of a square lattice, and (2) linear deformations of such media. Our bulk Hamiltonians break time-reversal symmetry due to perturbation by a magnetic term, which opens a band gap about the band structure degeneracies of the unperturbed bulk Hamiltonian. In case (1), these are quadratic band degeneracies; in case (2), they are pairs of conical degeneracies. We demonstrate that this band gap is traversed by two distinct edge state curves, consistent with the bulk-edge correspondence principle of topological physics. Blow-ups of these curves near the bulk band degeneracies are described by effective (homogenized) edge Hamiltonians derived via multiple-scale analysis which control the bifurcation of edge states. In case (1), the bifurcation is governed by a matrix Schrödinger operator; in case (2), it is governed by a pair of Dirac operators. We present analytical results and numerical simulations for both the full 2D edge Hamiltonian spectral problem and the spectra of effective edge Hamiltonians.

We study nonlinear bound states -- time-harmonic and spatially decaying ($L^2$) solutions -- of the nonlinear Schrödinger / Gross--Pitaevskii equations (NLS/GP) with a compactly supported linear potential. Such solutions are known to bifurcate from the $L^2$ bound states of an underlying Schrödinger operator $H_V=-\partial_x^2+V$. In this article we prove an extension of this result: for the 1D NLS/GP, nonlinear bound states also arise via bifurcation from the scattering resonance states and transmission resonance states of $H_V$, associated with the poles and zeros, respectively, of the reflection coefficients, $r_\pm(k)$, of $H_V$. The corresponding resonance states are non-decaying and only $L^2_{\rm loc}$. In contrast to nonlinear states arising from $L^2$ bound states of $H_V$, these resonance bifurcations initiate at a strictly positive $L^2$ threshold which is determined by the position of the complex scattering resonance pole or transmission resonance zero.

Consider the Schro"\dinger operator H =- \Delta + V, where the potential V is real, Z2-periodic, and additionally invariant under the symmetry group of the square. We show that, under typical small linear deformations of V, the quadratic band degeneracy points occurring over the high-symmetry quasimomentum M (see [R. T. Keller, J. L. Marzuola, B. Osting, and M. I. Weinstein, Multiscale Model. Simul., 16 (2018), pp. 1684--1731; R. T. Keller et al., Multiscale Model. Simul., 18 (2020), pp. 1371-1373]) each split into two separated degeneracies over perturbed quasimomenta D+ and D-, and that these degeneracies are Dirac points. The local character of the degenerate dispersion surfaces about the emergent Dirac points are tilted, elliptical cones. Correspondingly, the dynamics of wavepackets spectrally localized near either D+ or D-are governed by a system of Dirac equations with an advection term. Symmetry-breaking perturbations and induced band topology are also discussed.

There is increased interest in time-dependent (non-autonomous) Hamiltonians, stemming in part from the active field of Floquet quantum materials. Despite this, dispersive time-decay bounds, which reflect energy transport in such systems, have received little attention. We study the dynamics of non-autonomous, time-periodically forced, Dirac Hamiltonians: i partial derivative t alpha = /D(t)alpha, where /D(t) = i sigma 3 partial derivative x + nu(t) is time-periodic but not spatially localized. For the special case nu(t) = m sigma 1, which models a relativistic particle of constant mass m, one has a dispersive decay bound: ||alpha(t, x)||L infinity less than or similar to t-1 2 . Previous analyses of Schr & ouml;dinger Hamiltonians (e.g. [4,29,30,45]) suggest that this x decay bound persists for small, spatially-localized and time-periodic nu(t). However, we show that this is not necessarily the case if nu(t) is not spatially localized. Specifically, we study two non-autonomous Dirac models whose time-evolution (and monodromy operator) is constructed via Fourier analysis. In a rotating mass model, the dispersive decay bound is of the same type as for the constant mass model. However, in a model with a periodically alternating sign of the mass, the results are quite different. By stationary-phase analysis of the associated Fourier representation, we display initial data for which the L infinity x time-decay rate are considerably slower: O(t-1/3) or even O(t-1/5) as t -> infinity. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

We study a semilinear hyperbolic system of PDEs which arises as a continuum approximation of the discrete nonlinear dimer array model introduced by Hadad et al. (ACS Photonics 4:1974–1979, 2017). We classify the system’s traveling waves, and study their stability properties. We focus on traveling pulse solutions (“solitons”) on a nontrivial background and moving domain wall solutions (kinks); both arise as heteroclinic connections between spatially uniform equilibria of a reduced dynamical system. We present analytical results on: nonlinear stability and spectral stability of supersonic pulses, and spectral stability of moving domain walls. Our stability results are in terms of weighted H^1 norms of the perturbation, which capture the phenomenon of convective stabilization; as time advances, the traveling wave “outruns” the growing disturbance excited by an initial perturbation; the nontrivial spatially uniform equilibria are linearly exponentially unstable. We use our analytical results to interpret phenomena observed in numerical simulations.

We present lower bounds on tunneling rates in magnetic double well systems for generic values of the coupling constant. This result was recently announced in and complements our recent counter-example construction which exhibits vanishing tunneling for specially-constructed double-well potentials.

We revisit the problem of quantum tunneling for a particle moving in the continuum, and in the absence of a magnetic field. In all spatial dimensions, we extend previous results to the case where the single-well potential satisfies reflection-symmetry.

We present a magnetic double-well Hamiltonian where the tunneling between the two wells vanishes, as recently announced.

We study the dynamics of a gas bubble in a fluid with surface tension, initially near a spherical equilibrium. While there are many studies and applications of radial bubble dynamics, the theory of general deformations from a spherical equilibrium is less developed. We aim to understand how asymmetrically perturbed equilibrium bubbles evolve toward spherical equilibrium due to thermal or viscous dissipation in an incompressible liquid. We focus on the isobaric approximation [Prosperetti, JFM, 1991], under which the gas pressure within the bubble is spatially uniform and obeys the ideal gas law. The liquid outside the bubble is incompressible, irrotational, and has surface tension. We prove that any equilibrium gas bubble must be spherical by showing that the bubble boundary is a closed surface of constant mean curvature. We then study the initial value problem (IVP) for the coupled PDEs, constitutive laws and interface conditions of the isobaric approximation for general (asymmetric) small initial perturbations of the spherical bubble in the linearized approximation. Our first result, considering thermal damping without viscosity, proves that the linearized IVP is globally well-posed. The monopole (radial) component of the perturbation decays exponentially over time, while the multipole (non-radial) components undergo undamped oscillations. This indicates a limitation of the isobaric model for non-spherical dynamics. Our second result, incorporating viscous dissipation, shows that the IVP is linear and nonlinearly ill-posed due to an incompatibility of normal stress boundary conditions, for non-spherical solutions, and the irrotationality assumption. Our study concludes that to accurately capture the dynamics of general deformations of a gas bubble, the model must account for either vorticity generated at the bubble-fluid boundary, spatial non-uniformities in the gas pressure, or both.

Dispersive time-decay estimates are proved for a one-parameter family of onedimensional Dirac Hamiltonians with dislocations; these are operators which interpolate between two phase-shifted massive Dirac Hamiltonians at x = +\infty and x =-\infty. This family of Hamiltonians arises in the theory of topologically protected states of one-dimensional quantum materials. For certain values of the phase-shift parameter, \tau , the Dirac Hamiltonian has a threshold resonance at the endpoint of its essential spectrum. Such resonances are known to influence the time-decay rate. Our main result explicitly displays the transition in time-decay rate as \tau varies between resonant and nonresonant values. Our results appear to be the first dispersive time-decay estimates for Dirac Hamiltonians which are not a relatively compact perturbation of a free Dirac operator.