Surface bound states in the continuum

We introduce a novel concept of surface bound states in the continuum, i.e., surface modes embedded into the linear spectral band of a discrete lattice. We suggest an efficient method for creating such surface modes and the local bounded potential necessary to support the embedded modes. We demonstrate that the surface embedded modes are structurally stable, and the position of their eigenvalues inside the spectral band can be tuned continuously by adding weak nonlinearity.     

Scattering of long-wavelength acoustic phonons by isotopic impurities

We consider the problem of the relaxation of an arbitrary initial distribution function of a gas of long-wave acoustic phonons scattered by isotopic impurities embedded in a crystalline medium with cubic symmetry. The spectral decomposition of the collision integral of the suitable Boltzmann-Peierls equation is obtained. The spectrum of the collision operator is purely discrete and in addition to the eigenvalue 0 consists of three other eigenvalues. Explicit analytic expressions for these eigenvalues are obtained. Within the Chapman-Enskog approximation we derive the diffusion equation for the density of phonons and obtain the explicit expression for the diffusion coefficient. The dependency of the eigenvalues of the collision operator and the diffusion coefficient on the elastic constants of the medium is studied.     

Multiscale finite element methods for high-contrast problems using local spectral basis functions

In this paper we study multiscale finite element methods (MsFEMs) using spectral multiscale basis functions that are designed for high-contrast problems. Multiscale basis functions are constructed using eigenvectors of a carefully selected local spectral problem. This local spectral problem strongly depends on the choice of initial partition of unity functions. The resulting space enriches the initial multiscale space using eigenvectors of local spectral problem. The eigenvectors corresponding to small, asymptotically vanishing, eigenvalues detect important features of the solutions that are not captured by initial multiscale basis functions. Multiscale basis functions are constructed such that they span these eigenfunctions that correspond to small, asymptotically vanishing, eigenvalues. We present a convergence study that shows that the convergence rate (in energy norm) is proportional to (H/Λ₁)^1/2, where Λ₁ is proportional to the minimum of the eigenvalues that the corresponding eigenvectors are not included in the coarse space. Thus, we would like to reach to a larger eigenvalue with a smaller coarse space. This is accomplished with a careful choice of initial multiscale basis functions and the setup of the eigenvalue problems. Numerical results are presented to back-up our theoretical results and to show higher accuracy of MsFEMs with spectral multiscale basis functions. We also present a hierarchical construction of the eigenvectors that provides CPU savings.     

Comparing the structure of an emerging market with a mature one under global perturbation

In this paper we investigate the Tehran stock exchange (TSE) and Dow Jones Industrial Average (DJIA) in terms of perturbed correlation matrices. To perturb a stock market, there are two methods, namely local and global perturbation. In the local method, we replace a correlation coefficient of the cross-correlation matrix with one calculated from two Gaussian-distributed time series, whereas in the global method, we reconstruct the correlation matrix after replacing the original return series with Gaussian-distributed time series. The local perturbation is just a technical study. We analyze these markets through two statistical approaches, random matrix theory (RMT) and the correlation coefficient distribution. By using RMT, we find that the largest eigenvalue is an influence that is common to all stocks and this eigenvalue has a peak during financial shocks. We find there are a few correlated stocks that make the essential robustness of the stock market but we see that by replacing these return time series with Gaussian-distributed time series, the mean values of correlation coefficients, the largest eigenvalues of the stock markets and the fraction of eigenvalues that deviate from the RMT prediction fall sharply in both markets. By comparing these two markets, we can see that the DJIA is more sensitive to global perturbations. These findings are crucial for risk management and portfolio selection.     

Reduced nonlocal integrable mKdV equations of type (−λ, λ) and their exact soliton solutions

We conduct two group reductions of the Ablowitz–Kaup–Newell–Segur matrix spectral problems to present a class of novel reduced nonlocal reverse-spacetime integrable modified Korteweg–de Vries equations. One reduction is local, replacing the spectral parameter with its negative and the other is nonlocal, replacing the spectral parameter with itself. Then by taking advantage of distribution of eigenvalues, we generate soliton solutions from the reflectionless Riemann–Hilbert problems, where eigenvalues could equal adjoint eigenvalues.     

Cantor and Band Spectra for Periodic Quantum Graphs with Magnetic Fields

We provide an exhaustive spectral analysis of the two-dimensional periodic square graph lattice with a magnetic field. We show that the spectrum consists of the Dirichlet eigenvalues of the edges and of the preimage of the spectrum of a certain discrete operator under the discriminant (Lyapunov function) of a suitable Kronig-Penney Hamiltonian. In particular, between any two Dirichlet eigenvalues the spectrum is a Cantor set for an irrational flux, and is absolutely continuous and has a band structure for a rational flux. The Dirichlet eigenvalues can be isolated or embedded, subject to the choice of parameters. Conditions for both possibilities are given. We show that generically there are infinitely many gaps in the spectrum, and the Bethe-Sommerfeld conjecture fails in this case.The work was supported by the Deutsche Forschungsgemeinschaft, the Sonderforschungsbereich “Raum, Zeit, Materie” (SFB 647), and the International Bureau of BMBF at the German Aerospace Center (IB DLR, cooperation Germany–New Zealand NZL 05/001)     

Particle-hole asymmetry in doped mott insulators: implications for tunneling and photoemission spectroscopies

We use exact sum rules for the one-particle spectral function to quantify the idea that it is more difficult to add an electron than to extract one in a system with strong local repulsion. Our results explain the striking asymmetry in the tunneling spectra of underdoped cuprates which increases with underdoping. We also propose a novel method, based on ratios of sum rules, to estimate local density variations in inhomogeneous materials. Using a variational approach, we show that the origin of the particle-hole asymmetry lies in the incoherent spectral weight.     

On the stability of periodic solutions of the generalized Benjamin-Bona-Mahony equation

We study the stability of a four-parameter family of spatially periodic traveling wave solutions of the generalized Benjamin-Bona-Mahony equation under two classes of perturbations: periodic perturbations with the same periodic structure as the underlying wave, and long wavelength localized perturbations. In particular, we derive necessary conditions for spectral instability under perturbation for both classes of perturbations by deriving appropriate asymptotic expansions of the periodic Evans function, and we outline a theory of nonlinear stability under periodic perturbations based on variational methods which effectively extends our periodic spectral stability results.     

Diophantine Tori and Weyl Laws for Non-selfadjoint Operators in Dimension Two

We study the distribution of eigenvalues for non-selfadjoint perturbations of selfadjoint semiclassical analytic pseudodifferential operators in dimension two, assuming that the classical flow of the unperturbed part is completely integrable. An asymptotic formula of Weyl type for the number of eigenvalues in a spectral band, bounded from above and from below by levels corresponding to Diophantine invariant Lagrangian tori, is established. The Weyl law is given in terms of the long time averages of the leading non-selfadjoint perturbation along the classical flow of the unperturbed part.     

Eigenfunctions and Eigenvalues for a Scalar Riemann–Hilbert Problem Associated to Inverse Scattering

A complete set of eigenfunctions is introduced within the Riemann–Hilbert formalism for spectral problems associated to some solvable nonlinear evolution equations. In particular, we consider the time-independent and time-dependent Schr?dinger problems which are related to the KdV and KPI equations possessing solitons and lumps, respectively. Non-standard scalar products, orthogonality and completeness relations are derived for these problems. The complete set of eigenfunctions is used for perturbation theory and bifurcation analysis of eigenvalues supported by the potentials under perturbations. We classify two different types of bifurcations of new eigenvalues and analyze their characteristic features. One type corresponds to thresholdless generation of solitons in the KdV equation, while the other predicts a threshold for generation of lumps in the KPI equation. Received: 17 December 1998/ Accepted: 21 July 1999     

Evaluation and properties of a model for the determination of internal electric fields in proteins from the Stark effect of spectral holes

In this paper, we further evaluate a model for the analysis of Stark-shifted spectral holes, which allows for the determination of electric fields generated at the site of probe molecules by their host environment. In contrast to the conventional treatment of the Stark effect, our model regards both the internal electric field and the externally applied electric field as perturbations to the Hamiltonian of the probe molecule. Diagonalization of the Hamiltonian and fitting to the Stark-shifted spectral holes allows for the extraction of the internal electric field. We apply this model to Stark-shifted spectral holes burnt in a biological system, namely, myoglobin. Determining internal electric fields at the active sites of biological systems is particularly interesting, as these fields may contribute substantially to the function of such systems. As input to the data analysis procedure, state energies, transition dipole moments, and the local field factor are needed. We demonstrate here how the number of electronic states, the choice of local field factor, and the energies of several excited states of the probe molecule affect the resulting internal electric fields.     

Remarks on the integrable systems associated with rank 2 perturbations

In a paper by Moser, a class of completely integrable systems associated with the rank 2 perturbations of a symmetrical matrixA is given in the case that all eigenvalues ofA are distinct. This problem is also discussed by Alder and van Moerbeke in terms of the Kac-Moody algebra. In this Letter, we prove that these systems are also completely integrable in the case thatA has multiple eigenvalues by use of the moment map and the isospectral deformations.     

Spectral Theory of Massless Pauli-Fierz Models

We study the spectral theory of massless Pauli-Fierz models using an extension of the Mourre method. We prove the local finiteness of point spectrum and a limiting absorption principle away from the eigenvalues for an arbitrary coupling constant. In addition we show that the expectation value of the number operator is finite on all eigenvectors.Supported by Carlsbergfondet     

On the modulation equations and stability of periodic generalized Korteweg–de Vries waves via Bloch decompositions

In this paper, we consider the relation between Evans-function-based approaches to the stability of periodic travelling waves and other theories based on long-wavelength asymptotics together with Bloch wave expansions. In previous work it was shown by rigorous Evans function calculations that the formal slow modulation approximation resulting in the linearized Whitham averaged system accurately describes the spectral stability to long-wavelength perturbations. To clarify the connection between Bloch-wave-based expansions and Evans-function-based approaches, we reproduce this result without reference to the Evans function by using direct Bloch expansion methods and spectral perturbation analysis. One of the novelties of this approach is that we are able to calculate the relevant Bloch waves explicitly for arbitrary finite-amplitude solutions. Furthermore, this approach has the advantage of being applicable in the more general multi-periodic setting where no conveniently computable Evans function has yet been devised.     

Lieb-Thirring Inequalities for Geometrically Induced Bound States

We prove new inequalities of the Lieb-Thirring type on the eigenvalues of Schrödinger operators in wave guides with local perturbations. The estimates are optimal in the weak-coupling case. To illustrate their applications, we consider, in particular, a straight strip and a straight circular tube with either mixed boundary conditions or boundary deformations.     

Thermodynamic Limit for a Spin Lattice

An integrable spin lattice is a higher dimensional generalization of integrable spin chains. In this paper we consider a special spin lattice related to quantum mechanical interpretation of the three-dimensional lattice model in statistical mechanics (Zamolodchikov and Baxter). The integrability means the existence of a set of mutually commuting operators expressed in the terms of local spin variables. The significant difference between spin chain and spin lattice is that the commuting set for the latter is produced by a transfer matrix with two equitable spectral parameters. There is a specific bilinear functional equation for the eigenvalues of this transfer matrix.The spin lattice is investigated in this paper in the limit when both sizes of the lattice tend to infinity. The limiting form of bilinear equation is derived. It allows to analyze the distributions of eigenvalues of the whole commuting set. The ground state distribution is obtained explicitly. A structure of excited states is discussed.     

Linearization of vector fields and embedding of diffeomorphisms in flows via Nash–Moser theorem

In the first part of this paper we give suitable spectral properties of the adjoint operators induced by appropriate perturbations of some hyperbolic linear vector fields. These properties are useful to prove general facts based on the Nash–Moser inverse function theorem. In the second part of this work we study circumstances where a global linearization of a vector field X$X$ in a real numerical space is feasible and where some diffeomorphisms which are close to exp(X)$exp\left(X\right)$ can be embedded in a flow.     

Probabilistic Weyl Laws for Quantized Tori

For the Toeplitz quantization of complex-valued functions on a 2n-dimensional torus we prove that the expected number of eigenvalues of small random perturbations of a quantized observable satisfies a natural Weyl law (1.3). In numerical experiments the same Weyl law also holds for “false” eigenvalues created by pseudospectral effects.     

Synchronizability of network ensembles with prescribed statistical properties

It has been shown that synchronizability of a network is determined by the local structure rather than the global properties. With the same global properties, networks may have very different synchronizability. In this paper, we numerically studied, through the spectral properties, the synchronizability of ensembles of networks with prescribed statistical properties. Given a degree sequence, it is found that the eigenvalues and eigenratios characterizing network synchronizability have well-defined distributions, and statistically, the networks with extremely poor synchronizability are rare. Moreover, we compared the synchronizability of three network ensembles that have the same nodes and average degree. Our work reveals that the synchronizability of a network can be significantly affected by the local pattern of connections, and the homogeneity of degree can greatly enhance network synchronizability for networks of a random nature.     

Geometrical aspects on the dark matter problem

In the present paper we apply Nash’s theory of perturbative geometry to the study of dark matter gravity in a higher-dimensional space–time. It is shown that the dark matter gravitational perturbations at local scale can be explained by the extrinsic curvature of the standard cosmology. In order to test our model, we use a spherically symmetric metric embedded in a five-dimensional bulk. As a result, considering a sample of 10 low surface brightness and 6 high surface brightness galaxies, we find a very good agreement with the observed rotation curves of smooth hybrid alpha-HI measurements.