Spectral Gap Research Articles

Discrete Breathers of Nonlinear Dimer Lattices: Bridging the Anti-continuous and Continuous Limits

In this work, we study the dynamics of an infinite array of nonlinear dimer oscillators which are linearly coupled as in the classical model of Su, Schrieffer and Heeger (SSH). The ratio of in-cell and out-of-cell couplings of the SSH model defines distinct $\textit{phases}$: topologically trivial and topologically non-trivial. We first consider the case of weak out-of-cell coupling, corresponding to the topologically trivial regime for linear SSH; for any prescribed isolated dimer frequency, $\omega_b$, which satisfies non-resonance and non-degeneracy assumptions, we prove that there are discrete breather solutions for sufficiently small values of the out-of-cell coupling parameter. These states are $2\pi/\omega_b$- periodic in time and exponentially localized in space. We then study the global continuation with respect to this coupling parameter. We first consider the case where $\omega_b$, the seeding discrete breather frequency, is in the (coupling dependent) phonon gap of the underlying linear infinite array. As the coupling is increased, the phonon gap decreases in width and tends to a point (at which the topological transition for linear SSH occurs). In this limit, the spatial scale of the discrete breather grows and its amplitude decreases, indicating the weakly nonlinear long wave regime. Asymptotic analysis shows that in this regime the discrete breather envelope is determined by a vector gap soliton of the limiting envelope equations. We use the envelope theory to describe discrete breathers for SSH- coupling parameters corresponding to topologically trivial and, by exploiting an emergent symmetry, topologically nontrivial regimes, when the spectral gap is small. Our asymptotic theory shows excellent agreement with extensive numerical simulations over a wide range of parameters.

The dependency of spectral gaps on the convergence of the inverse iteration for a nonlinear eigenvector problem

In this paper, we consider the generalized inverse iteration for computing ground states of the Gross–Pitaevskii eigenvector (GPE) problem. For that we prove explicit linear convergence rates that depend on the maximum eigenvalue in magnitude of a weighted linear eigenvalue problem. Furthermore, we show that this eigenvalue can be bounded by the first spectral gap of a linearized Gross–Pitaevskii operator, recovering the same rates as for linear eigenvector problems. With this we establish the first local convergence result for the basic inverse iteration for the GPE without damping. We also show how our findings directly generalize to extended inverse iterations, such as the Gradient Flow Discrete Normalized (GFDN) proposed in [W. Bao and Q. Du, Computing the ground state solution of Bose–Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput. 25 (2004) 1674–1697] or the damped inverse iteration suggested in [P. Henning and D. Peterseim, Sobolev gradient flow for the Gross–Pitaevskii eigenvalue problem: Global convergence and computational efficiency, SIAM J. Numer. Anal. 58 (2020) 1744–1772]. Our analysis also reveals why the inverse iteration for the GPE does not react favorably to spectral shifts. This empirical observation can now be explained with a blow-up of a weighting function that crucially contributes to the convergence rates. Our findings are illustrated by numerical experiments.

Local density of states fluctuations in a two-dimensional superconductor as a probe of quantum diffusion

The interplay of superconductivity and disorder generates a wealth of complex phenomena. In particular, the peculiar structure of diffusive electronic wavefunctions is predicted to increase the superconducting critical temperature in some range of disorder. In this work, we use an epitaxial monolayer of lead showing a simple band structure and homogenous structural disorder as a model system of a 2D superconductor in the weak-antilocalization regime. Then, we perform an extensive study of the emergent fluctuations of local density of states (LDOS) and spectral energy gap in this material and compare them with both analytical results and numerical solution of the attractive Hubbard model. We show that mesoscopic LDOS fluctuations allow to probe locally both the elastic and inelastic scattering rates which are notoriously difficult to measure in transport measurements.

Analyticity of parametric elliptic eigenvalue problems and applications to quasi-Monte Carlo methods

In the present paper, we study the analyticity of the leftmost eigenvalue of the linear elliptic partial differential operators with random coefficient and analyse the convergence rate of the quasi-Monte Carlo method for approximation of the expectation of this quantity. The random coefficient is assumed to be represented by an affine expansion a 0 ( x ) + ∑ j ∈ N y j a j ( x ) , where elements of the parameter vector y = ( y j ) j ∈ N ∈ U ∞ are independent and identically uniformly distributed on U := [ − 1 2 , 1 2 ] . Under the assumption ‖ ∑ j ∈ N ρ j | a j | ‖ L ∞ ( D ) < ∞ with some positive sequence ( ρ j ) j ∈ N ∈ ℓ p ( N ) for p ∈ ( 0 , 1 ] we show that for any y ∈ U ∞ , the elliptic partial differential operator has a countably infinite number of eigenvalues ( λ j ( y ) ) j ∈ N which can be ordered non-decreasingly. Moreover, the spectral gap λ 2 ( y ) − λ 1 ( y ) is uniformly positive in U ∞ . From this, we prove the holomorphic extension property of λ 1 ( y ) to a complex domain in C ∞ and estimate partial derivatives of λ 1 ( y ) with respect to the parameter y by using Cauchy's formula for analytic functions. Based on these bounds we prove the dimension-independent convergence rate of the quasi-Monte Carlo method to approximate the expectation of λ 1 ( y ) .

Stable isoperimetric ratios and the Hodge Laplacian of hyperbolic manifolds

AbstractWe show that for a closed hyperbolic 3‐manifold, the size of the first eigenvalue of the Hodge Laplacian acting on coexact 1‐forms is comparable to an isoperimetric ratio relating geodesic length and stable commutator length with comparison constants that depend polynomially on the volume and on a lower bound on injectivity radius, refining estimates of Lipnowski and Stern. We use this estimate to show that there exist sequences of closed hyperbolic 3‐manifolds with injectivity radius bounded below and volume going to infinity for which the 1‐form Laplacian has spectral gap vanishing exponentially fast in the volume.

Spectral properties of size-invariant shape transformation.

Size-invariant shape transformation is a technique of changing the shape of a domain while preserving its sizes under the Lebesgue measure. In quantum-confined systems, this transformation leads to so-called quantum shape effects in the physical properties of confined particles associated with the Dirichlet spectrum of the confining medium. Here we show that the geometric couplings between levels generated by the size-invariant shape transformations cause nonuniform scaling in the eigenspectra. In particular, the nonuniform level scaling, in the direction of increasing quantum shape effect, is characterized by two distinct spectral features: lowering of the first eigenvalue (ground-state reduction) and changing of the spectral gaps (energy level splitting or degeneracy formation depending on the symmetries). We explain the ground-state reduction by the increase in local breadth (i.e., parts of the domain becoming less confined) that is associated with the sphericity of these local portions of the domain. We accurately quantify the sphericity using two different measures: the radius of the inscribed n-sphere and the Hausdorff distance. Due to Rayleigh-Faber-Krahn inequality, the greater the sphericity, the lower the first eigenvalue. Then level splitting or degeneracy, depending on the symmetries of the initial configuration, becomes a direct consequence of size invariance dictating the eigenvalues to have the same asymptotic behavior due to Weyl law. Such level splittings may be interpreted as geometric analogs of Stark and Zeeman effects. Furthermore, we find that the ground-state reduction causes a quantum thermal avalanche which is the underlying reason for the peculiar effect of spontaneous transitions to lower entropy states in systems exhibiting the quantum shape effect. Unusual spectral characteristics of size-preserving transformations can assist in designing confinement geometries that could lead to classically inconceivable quantum thermal machines.

On the spectral gap of higher-dimensional Schrödinger operators on large domains

We study the asymptotic behaviour of the spectral gap of Schrödinger operators in two and higher dimensions and in a limit where the volume of the domain tends to infinity. Depending on properties of the underlying potential, we will find different asymptotic behaviours of the gap. In some cases the gap behaves as the gap of the free Dirichlet Laplacian and in some cases it does not.

Questions About Extreme Points

We discuss the geometry of the unit ball—specifically, the structure of its extreme points (if any)—in subspaces of L^1 and L^infty on the circle that are formed by functions with prescribed spectral gaps. A similar issue is considered for kernels of Toeplitz operators in H^infty .

Cross Spectral Disparity Estimation From VIS and NIR Paired Images Using Disentangled Representation and Reversible Neural Networks

There exists spectral gap between visible (VIS) and near infrared (NIR) images causing dissimilar intensity according to reflection property of objects and materials. Therefore, it has a limit of applying traditional stereo matching to cross spectral disparity estimation. In this paper, we propose cross spectral disparity estimation from VIS and NIR paired images using disentangled representation and reversible neural networks. We build a supervised learning framework based on reversible blocks to extract scene features robust against the spectral gap. Reversible blocks decompose features into scene and style components to bridge the spectral gap between VIS and NIR images. We perform stereo matching on the scene component to get an initial disparity map by a 3D convolutional neural network. To generate clear edges in the disparity map, we use a semantic segmentation network as auxiliary information to refine the initial disparity map. Besides, to consider the lack of the ground truth, we synthesize reference disparity maps with guided image filtering. Experimental results demonstrate that the proposed method achieves accurate edges in disparity along object boundaries and outperforms the state-of-the-art methods in both visual comparison and quantitative measurements.

Pathwise unstable invariant manifolds reduction for stochastic evolution equations driven by nonlinear noise

This paper is concerned with the pathwise dynamics of the stochastic evolution equation: du + Audt = F(u)dt + G(u)dW(t) on a separable Hilbert space H with the Lipschitz continuous drift term F(u) as well as the Lipschitz continuous diffusion term G(u). We first introduce the notion of generalized random dynamical systems (GRDSs) and show that the equation can generate a GRDS. We then construct a pathwise unstable manifold for the GRDS provided that the Lipschitz constants of the drift term and the diffusion term satisfy a spectral gap condition. At last, we present a pathwise unstable manifold reduction for the GRDS.

Impediments to diffusion in quantum graphs: Geometry-based upper bounds on the spectral gap

We derive several upper bounds on the spectral gap of the Laplacian on compact metric graphs with standard or Dirichlet vertex conditions. In particular, we obtain estimates based on the length of a shortest cycle (girth), diameter, total length of the graph, as well as further metric quantities introduced here for the first time, such as the avoidance diameter. Using known results about Ramanujan graphs, a class of expander graphs, we also prove that some of these metric quantities, or combinations thereof, do not to deliver any spectral bounds with the correct scaling.

Perturbations of periodic Sturm–Liouville operators

We study perturbations of the self-adjoint periodic Sturm–Liouville operatorA0=1r0(−ddxp0ddx+q0) and conclude under L1-assumptions on the differences of the coefficients that the essential spectrum and absolutely continuous spectrum remain the same. If a finite first moment condition holds for the differences of the coefficients, then at most finitely many eigenvalues appear in the spectral gaps. This observation extends a seminal result by Rofe-Beketov from the 1960s. Finally, imposing a second moment condition we show that the band edges are no eigenvalues of the perturbed operator.

Offset responses in primary auditory cortex are enhanced after notched noise stimulation.

Sensory "aftereffects" are a subgroup of sensory illusions that can be defined as an illusory phenomenon triggered after prolonged exposure to a given sensory inducer. These phenomena are interesting because they can provide insights into the mechanisms of perception. In auditory modality, there is a special interest in the so-called "Zwicker tone" (ZT), an auditory aftereffect triggered after the presentation of a notched noise (NN, broadband noise with a missing frequency band). The ZT has been considered a plausible model of a specific tinnitus subtype since it presents some key characteristics in common with tinnitus. Indeed, both the tinnitus percept and ZT can be triggered by a relative "sensory deprivation," and their pitch corresponds to the frequency region that has been sensory deprived. The effects of a NN presentation on the central auditory system are still barely investigated, and the mechanisms of the ZT are elusive. In this study, we analyzed the laminar structure of the neural activity in the primary cortex of anesthetized and awake guinea pigs during and after white noise (WN) and NN stimulation. We found significantly increased offset responses, in terms of both spiking activity and local field potential amplitude, after NN compared with WN presentation. The offset responses were circumscribed to the granular and upper infragranular layers (input layers) and were maximal when the neuron's best frequency was within or near the missing frequency band. The mechanisms of the offset response and its putative link with the ZT are discussed.NEW & NOTEWORTHY Notched noise (white noise with embedded spectral gap) causes significant excitatory offset responses in the auditory cortex of awake and anesthetized guinea pigs. The largest offset responses were located in the infragranular/granular layers, and current source density analysis revealed that offset responses were associated with an early current sink localized in the upper infragranular layers. We discuss the possibility that the offset responses might be associated with an auditory phantom percept (Zwicker tone).

The spectral gap of random regular graphs

AbstractWe bound the second eigenvalue of random ‐regular graphs, for a wide range of degrees , using a novel approach based on Fourier analysis. Let be a uniform random ‐regular graph on vertices, and be its second largest eigenvalue by absolute value. For some constant and any degree with , we show that asymptotically almost surely. Combined with earlier results that cover the case of sparse random graphs, this fully determines the asymptotic value of for all . To achieve this, we introduce new methods that use mechanisms from discrete Fourier analysis, and combine them with existing tools and estimates on ‐regular random graphs—especially those of Liebenau and Wormald.

A nonvanishing spectral gap for AKLT models on generalized decorated graphs

We consider the spectral gap question for Affleck, Kennedy, Lieb, and Tasaki models defined on decorated versions of simple, connected graphs G. This class of decorated graphs, which are defined by replacing all edges of G with a chain of n sites, in particular includes any decorated multi-dimensional lattice. Using the Tensor Network States approach from [Abdul-Rahman et al., Analytic Trends in Mathematical Physics, Contemporary Mathematics (American Mathematical Society, 2020), Vol. 741, p. 1.], we prove that if the decoration parameter is larger than a linear function of the maximal vertex degree, then the decorated model has a nonvanishing spectral gap above the ground state energy.

Mixing of the averaging process and its discrete dual on finite-dimensional geometries

We analyze the $L^1$-mixing of a generalization of the Averaging process introduced by Aldous. The process takes place on a growing sequence of graphs which we assume to be finite-dimensional, in the sense that the random walk on those geometries satisfies a family of Nash inequalities. As a byproduct of our analysis, we provide a complete picture of the total variation mixing of a discrete dual of the Averaging process, which we call Binomial Splitting process. A single particle of this process is essentially the random walk on the underlying graph. When several particles evolve together, they interact by synchronizing their jumps when placed on neighboring sites. We show that, given $k$ the number of particles and $n$ the (growing) size of the underlying graph, the system exhibits cutoff in total variation if $k\to\infty$ and $k=O(n^2)$. Finally, we exploit the duality between the two processes to show that the Binomial Splitting satisfies a version of Aldous' spectral gap identity, namely, the relaxation time of the process is independent of the number of particles.

Magnetic perturbations of the Robin Laplacian in the strong coupling limit

This paper is devoted to the asymptotic analysis of the eigenvalues of the Laplace operator with a strong magnetic field and Robin boundary condition on a smooth planar domain and with a negative boundary parameter. We study the singular limit when the Robin parameter tends to infinity, which is equivalent to a semi-classical limit involving a small positive semi-classical parameter. The main result is a comparison between the spectrum of the Robin Laplacian with an effective operator defined on the boundary of the domain via the Born–Oppenheimer approximation. More precisely, the low-lying eigenvalue of the Robin Laplacian is approximated by those of the effective operator. When the curvature has a unique non-degenerate maximum, we estimate the spectral gap and find that the magnetic field does not contribute to the three-term expansion of the eigenvalues. In the case of the disc domains, the eigenvalue asymptotics displays the contribution of the magnetic field explicitly.

Dirichlet Sub-Laplacians on Homogeneous Carnot Groups: Spectral Properties, Asymptotics, and Heat Content

Abstract We consider sub-Laplacians in open bounded sets in a homogeneous Carnot group and study their spectral properties. We prove that these operators have a pure point spectrum and prove the existence of the spectral gap. In addition, we give applications to the small ball problem for a hypoelliptic Brownian motion and the large time behavior of the heat content in a regular domain.

Inference on the maximal rank of time-varying covariance matrices using high-frequency data

We study the rank of the instantaneous or spot covariance matrix ΣX(t) of a multidimensional process X(t). Given high-frequency observations X(i/n), i=0,…,n, we test the null hypothesis rank(ΣX(t))≤r for all t against local alternatives where the average (r+1)st eigenvalue is larger than some signal detection rate vn. A major problem is that the inherent averaging in local covariance statistics produces a bias that distorts the rank statistics. We show that the bias depends on the regularity and spectral gap of ΣX(t). We establish explicit matrix perturbation and concentration results that provide nonasymptotic uniform critical values and optimal signal detection rates vn. This leads to a rank estimation method via sequential testing. For a class of stochastic volatility models, we determine data-driven critical values via normed p-variations of estimated local covariance matrices. The methods are illustrated by simulations and an application to high-frequency data of U.S. government bonds.

Cooper Pair Excitation Mediated by a Molecular Quantum Spin on a Superconducting Proximitized Gold Film.

Breaking a correlated pair in a superconductor requires an even number of fermions providing at least twice the pairing energy Δ. Here, we show that a single tunneling electron can also excite a pair breaking excitation in a proximitized gold film in the presence of magnetic impurities. Combining scanning tunneling spectroscopy with theoretical modeling, we map the excitation spectrum of an Fe-porphyrin molecule on the Au/V(100) proximitized surface into a manifold of entangled Yu-Shiba-Rusinov and spin excitations. Pair excitations emerge in the tunneling spectra as peaks outside the spectral gap only in the strong coupling regime, where the presence of a bound quasiparticle in the ground state ensures the even fermion parity of the excitation. Our results unravel the quantum nature of magnetic impurities on superconductors and demonstrate that pair excitations unequivocally reveal the parity of the ground state.