Singular Continuous Spectrum Research Articles

Non-compact Quantum Graphs with Summable Matrix Potentials

Let $\mathcal{G}$ be a metric noncompact connected graph with finitely many edges. The main object of the paper is the Hamiltonian ${\bf H}_{\alpha}$ associated in $L^2(\mathcal{G};\mathbb{C}^m)$ with a matrix Sturm-Liouville expression and boundary delta-type conditions at each vertex. Assuming that the potential matrix is summable and applying the technique of boundary triplets and the corresponding Weyl functions, we show that the singular continuous spectrum of the Hamiltonian ${\bf H}_{\alpha}$ as well as any other self-adjoint realization of the Sturm-Liouville expression is empty. We also indicate conditions on the graph ensuring pure absolute continuity of the positive part of ${\bf H}_{\alpha}$. Under an additional condition on the potential matrix, a Bargmann-type estimate for the number of negative eigenvalues of ${\bf H}_{\alpha}$ is obtained. Additionally, for a star graph $\mathcal{G}$ a formula is found for the scattering matrix of the pair $\{{\bf H}_{\alpha}, {\bf H}_D\}$, where ${\bf H}_D$ is the Dirichlet operator on $\mathcal{G}$.

Plasmonic eigenvalue problem for corners: Limiting absorption principle and absolute continuity in the essential spectrum

We consider the plasmonic eigenvalue problem for a general 2D domain with a curvilinear corner, studying the spectral theory of the Neumann–Poincaré operator of the boundary. A limiting absorption principle is proved, valid when the spectral parameter approaches the essential spectrum. Putting the principle into use, it is proved that the corner produces absolutely continuous spectrum of multiplicity 1. The embedded eigenvalues are discrete. In particular, there is no singular continuous spectrum.

Periodic Jacobi matrices on trees

We begin the systematic study of the spectral theory of periodic Jacobi matrices on trees including a formal definition. The most significant result that appears here for the first time is that these operators have no singular continuous spectrum. We review important previous results of Sunada and Aomoto and present several illuminating examples. We present many open problems and conjectures that we hope will stimulate further work.

Noncompact complete Riemannian manifolds with singular continuous spectrum embedded into the essential spectrum of the Laplacian, I. The hyperbolic case

We construct Riemannian manifolds with singular continuous spectrum embedded in the absolutely continuous spectrum of the Laplacian. Our manifolds are asymptotically hyperbolic with sharp curvature bounds.

One-dimensional Discrete Dirac Operators in a Decaying Random Potential I: Spectrum and Dynamics

We study the spectrum and dynamics of a one-dimensional discrete Dirac operator in a random potential obtained by damping an i.i.d. environment with an envelope of type $n^{-\alpha}$ for $\alpha>0$. We recover all the spectral regimes previously obtained for the analogue Anderson model in a random decaying potential, namely: absolutely continuous spectrum in the super-critical region $\alpha>\frac12$; a transition from pure point to singular continuous spectrum in the critical region $\alpha=\frac12$; and pure point spectrum in the sub-critical region $\alpha<\frac12$. From the dynamical point of view, delocalization in the super-critical region follows from the RAGE theorem. In the critical region, we exhibit a simple argument based on lower bounds on eigenfunctions showing that no dynamical localization can occur even in the presence of point spectrum. Finally, we show dynamical localization in the sub-critical region by means of the fractional moments method and provide control on the eigenfunctions.

Emergence of anomalous dynamics from the underlying singular continuous spectrum in interacting many-body systems

We investigate the dynamical properties of an interacting many-body system with a non-trivial energy potential landscape that may induce a singular continuous single-particle energy spectrum. Focusing on the Aubry-Andr\'e model, whose anomalous transport properties in presence of interaction has recently been demonstrated experimentally in an ultracold gas setup, we discuss the anomalous slowing down of the dynamics it exhibits and show that it emerges from the singular-continuous nature of the single-particle excitation spectrum. Our study demonstrates that singular-continuous spectra can be found in interacting systems, unlike previously conjectured by treating the interactions in the mean-field approximation. This, in turns, also highlights the importance of the many-body correlations in giving rise to anomalous dynamics, which, in many-body systems, can result from a non-trivial interplay between geometry and interactions.

Absolute continuity of the spectrum in a twisted Dirichlet-Neumann waveguide

Quantum waveguide with the shape of planar infinite straight strip and combined Dirichlet and Neumann boundary conditions on the opposite half-lines of the boundary is considered. The absence of the point as well as of the singular continuous spectrum is proved.

Quantum graphs with summable matrix potentials

Let G be a metric, finite, noncompact, and connected graph with finitely many edges and vertices. Assume also that the length at least of one of the edges is infinite. The main object of the paper is Hamiltonian Hα associated in L2(G; Cm) with matrix Sturm-Liouville’s expression and boundary delta-type conditions at each vertex. Assuming that the potential matrix is summable and applying technique of boundary triplets and the corresponding Weyl functions we show that the singular continuous spectrum of the Hamiltonian Hα as well as any other self-adjoint realization of the Sturm-Liouville expression is empty. We also indicate conditions on the graph ensuring the positive part of the Hamiltonian Hα to be purely absolutely continuous. Under an additional condition on the potential matrix the Bargmann type estimate for the number of the negative eigenvalues of the operator Hα is obtained. Also we find a formula for the scattering matrix of the pair {Hα, HD} where HD is the operator of the Dirichlet problem on the graph.

Quantum Graphs with Summable Matrix Potentials

Let $$\mathcal{G}$$ be a metric, finite, noncompact, and connected graph with finitely many edges and vertices. Assume that the length of at least one of the edges is infinite. The main object of this paper is the Hamiltonian $${{{\mathbf{H}}}_{\alpha }}$$ associated in $${{L}^{2}}(\mathcal{G};{{\mathbb{C}}^{m}})$$ with a matrix Sturm–Liouville expression and boundary delta-type conditions at each vertex. Assuming that the potential matrix is summable and applying the technique of boundary triplets and the corresponding Weyl functions, we show that the singular continuous spectrum of the Hamiltonian $${{{\mathbf{H}}}_{\alpha }}$$ and any other self-adjoint realization of the Sturm–Liouville expression is empty. We also indicate conditions on the graph ensuring that the positive part of $${\mathbf{H}_{\alpha }}$$ is purely absolutely continuous. Under an additional condition on the potential matrix, a Bargmann-type estimate for the number of negative eigenvalues of the operator $${{{\mathbf{H}}}_{\alpha }}$$ is obtained. Additionally, a formula is found for the scattering matrix of the pair $$\{ {{{\mathbf{H}}}_{\alpha }},{{{\mathbf{H}}}_{D}}\} $$, where $${{{\mathbf{H}}}_{D}}$$ is the operator of the Dirichlet problem on the graph.

On singular spectrum of finite dimensional perturbations (to the Aronszajn-Donoghue-Kac theory)

The main results of the Aronszajn-Donoghue-Kac theory are extended to the case of n-dimensional (in the resolvent sense) perturbations à of an operator A0 = A0* defined on a Hilbert space H. Applying technique of boundary triplets we describe singular continuous and point spectra of extensions AB of a symmetric operator A acting in H in terms of the Weyl function M(·) of the pair {A, A0} and boundary n-dimensional operator B = B*. Assuming that the multiplicity of singular spectrum of A0 is maximal it is established orthogonality of singular parts EsAв and EsAo of the spectral measures EAв and EAo of the operators AB and A0, respectively. It is shown that the multiplicity of singular spectrum of special extensions of direct sums A = A(1) ⊕ A(2) cannot be maximal as distinguished from multiplicity of the absolutely continuous spectrum. In particular, it is obtained a generalization of the Kac theorem on multiplicity of singular spectrum of Schrodinger operator on the line as well as its clarification. The multiplicity of singular spectrum of special extensions of direct sums A = A(1) ⊕ A(2) are investigated. In particular, it is shown that it cannot be maximal as distinguished from multiplicity of the absolutely continuous spectrum. This result generalizes the Kac theorem on multiplicity of singular spectrum of Schrodinger operator on the line and clarifies it.

On Singular Spectrum of Finite-Dimensional Perturbations (toward the Aronszajn–Donoghue–Kac Theory)

The main results of the Aronszajn–Donoghue–Kac theory are extended to the case of n-dimensional (in the resolvent sense) perturbations $$\tilde {A}$$ of an operator $${{A}_{0}} = A_{0}^{ * }$$ defined on a Hilbert space $$\mathfrak{H}$$ . By applying the technique of boundary triplets, the singular continuous and point spectra of extensions AB of a symmetric operator A are described in terms of the Weyl function $$M( \cdot )$$ of the pair {A, A0} and an n-dimensional boundary operator B = B*. Assuming that the multiplicity of the singular spectrum of A0 is maximal, we establish the orthogonality of the singular parts $$E_{{{{A}_{B}}}}^{s}$$ and $$E_{{{{A}_{0}}}}^{s}$$ of the spectral measure $${{E}_{{{{A}_{B}}}}}$$ and $${{E}_{{{{A}_{0}}}}}$$ of the operators AB and A0, respectively. The multiplicity of the singular spectrum of special extensions of direct sums $$A = {{A}^{{(1)}}} \oplus {{A}^{{(2)}}}$$ is investigated. In particular, it is shown that this multiplicity cannot be maximal, as distinguished from the multiplicity of the absolutely continuous spectrum. This result generalizes and refines the Kac theorem on the multiplicity of the singular spectrum of the Schrodinger operator on the line.

Strange nonchaotic attractors in a nonsmooth dynamical system

Strange nonchaotic attractors (SNAs) have fractal geometric structure, but are nonchaotic in the dynamical sense. Since Grebogi et al. discovered SNA in 1984, it has become one of the important topics in nonlinear dynamics. Now, the study of SNAs has been mainly confined to smooth dynamics with quasiperiodic excitation or random excitation. In this paper, we consider a class of single-degree-of-freedom gear dynamical system with quasiperiodic forcing. We show that the gear transmission system can be modeled as a three-dimensional piecewise linear system, which belongs to a typical class of nonsmooth system. We then show that SNAs do exist in such nonsmooth dynamical system with quasiperiodic force. The dynamical behavior of the nonsmooth system is analyzed as a parameter is varied. The dynamics is analyzed through phase diagrams and bifurcation diagrams, Lyapunov exponents, singular continuous power spectrum, phase sensitivity of time series and rational approximations.

Singular Continuous Spectrum and Generic Full Spectral/Packing Dimension for Unbounded Quasiperiodic Schrödinger Operators

We proved that Schr\"odinger operators with unbounded potentials $(H_{\alpha,\theta}u)_n=u_{n+1}+u_{n-1}+ \frac{g(\theta+n\alpha)}{f(\theta+n\alpha)} u_n$ have purely singular continuous spectrum on the set $\{E: 0<L(E)<\delta{(\alpha,\theta;f,g)}\}$, where $\delta$ is an explicit function and $L$ is the Lyapunov exponent. We only require $f,g$ are H\"older continuous functions and $f$ has finitely many zeros with weak non-degenerate assumptions. Moreover, we show that for generic $\alpha$ and a.e. $\theta$, the spectral measure of $H_{\alpha,\theta}$ has full spectral/packing dimension.

Invariants of the trace map and uniform spectral properties for discrete Sturmian Dirac operators

We establish invariants for the trace map associated to a family of 1D discrete Dirac operators with Sturmian potentials. Using these invariants we prove that the operators have purely singular continuous spectrum of zero Lebesgue measure, uniformly on the mass and parameters that define the potentials. For rotation numbers of bounded density we prove that these Dirac operators have purely $\alpha$-continuous spectrum, as to the Schrodinger case, for some $\alpha \in (0,1)$. To the Sturmian Schrodinger and Dirac models we establish a comparison between invariants of the trace maps, which allows to compare the numbers $\alpha$'s and lower bounds on transport exponents.

Mechanisms of strange nonchaotic attractors in a nonsmooth system with border-collision bifurcations

It is not very clear to understand genesis and mechanisms for the creation of strange nonchaotic attractors (SNAs) due to the nonsmooth bifurcations in the nonsmooth systems. A quasiperiodically forced piecewise Logistic system is shown to exhibit many types of routes to the creation of SNAs. We point out that the truncation of border-collision torus-doubling bifurcation can lead to different types of SNAs. We identify and describe the Heagy–Hammel routes, fractalization route and intermittent routes after the two coexisting tori collide at the border and the doubled torus is interrupted in this system. It has been shown that there exist two critical tongue-type regions in the parameter space, where the different mechanisms for the birth of SNAs are investigated. These SNAs are identified by the Lyapunov exponents and the phase sensitivity exponents. Different types of SNAs are also characterized by the singular-continuous spectrum, Fourier transform, rational approximations, distribution of finite-time Lyapunov exponents and recurrence analysis.

Hyperuniformity and anti-hyperuniformity in one-dimensional substitution tilings.

This work considers the scaling properties characterizing the hyperuniformity (or anti-hyperuniformity) of long-wavelength fluctuations in a broad class of one-dimensional substitution tilings. A simple argument is presented which predicts the exponent α governing the scaling of Fourier intensities at small wavenumbers, tilings with α > 0 being hyperuniform, and numerical computations confirm that the predictions are accurate for quasiperiodic tilings, tilings with singular continuous spectra and limit-periodic tilings. Quasiperiodic or singular continuous cases can be constructed with α arbitrarily close to any given value between -1 and 3. Limit-periodic tilings can be constructed with α between -1 and 1 or with Fourier intensities that approach zero faster than any power law.

Absence of singular continuous spectrum for perturbed discrete Schrödinger operators

We show that the spectral measure of discrete Schr\"odinger operators $ (Hu)(n)= u({n+1})+u({n-1})+V(n)u(n)$ does not have singular continuous component if the potential $V(n)=O(n^{-1})$.

Level repulsion for Schrödinger operators with singular continuous spectrum

We describe a family of half-line continuum Schrödinger operators with purely singular continuous spectrum on (0, \infty) , exhibiting asymptotic strong level repulsion (known as clock behavior). This follows from the convergence of the renormalized continuum Christoffel–Darboux kernel to the sine kernel.

Strange nonchaotic dynamics of parametrically enhanced MLC circuit

We report experimental and numerical studies of a strange nonchaotic attractor (SNA) using a parametrically perturbed Murali–Lakshmanan–Chua (MLC) circuit. One of the energy storage elements in the circuit, i.e., the capacitor, is perturbed parametrically with a sinusoidal excitation. Under this condition, the circuit exhibits both chaotic and strange nonchaotic behavior for suitably adjusted capacitor values. This constitutes the novelty of this work. The other point to note in this circuit is that one of the two alternating-current (AC) stimulations (which are usually connected in series to observe SNA) is obtained by varying the capacitance with a perturbation consisting of a periodic stimulating force. The experimental results are confirmed by numerical study, in particular calculations of the correlation coefficient, path of translation variables, modified mean square displacement, singular continuous spectrum, power spectrum, and autocorrelation coefficient, to confirm the existence of an SNA in the dynamics of the modified MLC circuit.

Spectral transition line for the extended Harper's model in the positive Lyapunov exponent regime

We study the spectral transition line of the extended Harper's model in the positive Lyapunov exponent regime. We show that both pure point spectrum and purely singular continuous spectrum occur for dense subsets of frequencies on the transition line.