Localized Eigenfunctions Research Articles

Selfdual Einstein Metrics with Torus Symmetry

It is well-known that any 4-dimensional hyperkähler metric with two commuting Killing fields may be obtained explicitly, via the Gibbons-Hawking Ansatz, from a harmonic function invariant under a Killing field on • 3. In this paper, we find all selfdual Einstein metrics of nonzero scalar curvature with two commuting Killing fields. They are given explicitly in terms of a local eigenfunction of the Laplacian on the hyperbolic plane. We discuss the relation of this construction to a class of selfdual spaces found by Joyce, and some Einstein-Weyl spaces found by Ward, and then show that certain 'multipole' hyperbolic eigenfunctions yield explicit formulae for the quaternion-kähler quotients of • Pm—1 by an (m — 2)-torus studied by Galicki and Lawson. As a consequence we are able to place the well-known cohomogeneity one metrics, the quaternion-kähler quotients of • P2 (and noncompact analogues), and the more recently studied selfdual Einstein Hermitian metrics in a unified framework, and give new complete examples.

The kerzman-stein theorem on the sphere

Spherical monogenics of complex degree correspond to local eigenfunctions of the (Atiyah-Singer) Dirac operator on the unit sphere. Sm−1 of . In this paper we will consider the L2 -boundary value theory for this class of functions. The main Theorem is a higher dimensional version of the Kerzman-Stein Theorem of complex analysis in the plane.

Local orthogonal transformation and one-way methods for acoustic waveguides

A numerical method is developed for solving the two-dimensional Helmholtz equation in a region bounded by a flat top and a curved bottom. A local orthogonal transformation is first used to flatten the curved bottom of the waveguide. The one-way re-formulation based on the Dirichlet-to-Neumann map is then used to reduce the boundary value problem to an initial value problem. Numerical implementation of the resulting operator Riccati equation uses a large range step method for discretizing the range variable and a truncated local eigenfunction expansion for approximating the operators. This method is particularly useful for solving long range wave propagation problems in slowly varying waveguides.

Essential instabilities of fronts: bifurcation, and bifurcation failure

Various instability mechanisms of fronts in reaction-diffusion systems are analysed; the emphasis is on instabilities that have the potential to create modulated (i.e. time-periodic) waves near the primary front. Hopf bifurcations caused by point spectrum with associated localized eigenfunctions provide an example of such an instability. A different kind of instability occurs if one of the asymptotic rest states destabilizes: these instabilities are caused by essential spectrum. It is demonstrated that, if the rest state ahead of the front destabilizes, then modulated fronts are created that connect the rest state behind the front with small spatially periodic patterns ahead of the front. These modulated fronts are stable provided the spatially periodic patterns are stable. If, on the other hand, the rest state behind the front destabilizes, then modulated fronts that leave a spatially periodic pattern behind do not exist.

Highly excited vibronic eigenfunctions in a multimode nonadiabatic system with Duschinsky rotation

We study the characteristics of vibronic eigenfunctions of a multidimensional nonadiabatic system and their consequences in the quantum spectra. As an illustrative example, we investigate the properties of highly excited eigenfunctions of Heller’s multimode nonadiabatic system. The system consists of two diabatic states and two-dimensional (two-mode) harmonic potentials that are nonadiabatically coupled with the Condon approximation and with an appropriate magnitude of the Duschinsky angle. “Quantum chaos” thus produced has no classical counterpart. In addition to rather characterless chaotic eigenfunctions that are uniformly widespread in configuration space, we have found highly excited localized eigenfunctions of two extreme types which favor either the diabatic picture or adiabatic picture. As a result, the features of the associated quantum spectra are strongly affected by the initial preparation of a wave packet. This finding suggests that one can control the rate of nonadiabatic transitions such as that for electron transfer by using laser techniques or by choosing appropriate solvents.

LIFSHITZ ASYMPTOTICS AND LOCALIZATION FOR RANDOM QUANTUM WAVEGUIDES

We investigate a family of Dirichlet Laplacians on randomly dented or bulged strips in ℝ2; for this random quantum waveguide model, dense point spectrum with exponentially localized eigenfunctions near its fluctuation boundary at the bottom of the spectrum and Lifshitz asymptotics of the integrated density of states are established. For this purpose, multi-scale analysis in the quite abstract form of [21] is applied, and domain perturbations of the Laplacian are studied.

A note on Rayleigh stability criterion for compressible flows

This paper examines a sufficient instability criterion at large wave numbers for inviscid compressible swirling basic flows depending only on the radial coordinate. The growth rate of these instabilities, obtained through a WKB high wave number analysis, is compared with a normal mode analysis. It is proved analytically that both growth rates match exactly in the limit of high wave numbers. Furthermore, the eigenvalues and localized eigenfunctions of the most amplified discrete modes are in agreement analytically and numerically.

Local eigenfunctions based suboptimal wavelet packet representation of contaminated chaotic signals

We report a suboptimal wavelet packet representation (SWPR) of signals emanating from a chaotic attractor contaminated by low levels of noise. Our method-geared towards choosing a suboptimal scaling function to parsimoniously represent the signal-involves extracting local eigenfunctions using artificial ensembles generated from a pseudo-probability space, and using the extracted local eigenfunctions to develop a suboptimal scaling function. The application of our novel representation method to actual acoustic emission (AE) signals, sampled as time-series data (TSD) from the turning process, reveals the superiority of these methods over the existing signal representations.

Taylor and laurent series on the sphere

Spherical monogenics of complex degree correspond to local eigenfunctions of the (Atiyah-Singer) Dirac operator on the unit sphere Sm-1 of R m. In this paper we will given explicit formulae for the Taylor and Laurent series for this class of functions. This leads to an explicity residue theory for spherical monogenics having point singularities.

Quadrilateral lattices and eigenfunction potentials for N-component KP hierarchies

The (charged) free fermion formulation of the N-component KP hierarchies is used for the description of quadrilateral lattices in terms of multicomponent KP eigenfunction potentials. Tau functions for these hierarchies are explicitly calculated, giving rise to Grammian and Casorati determinant solutions to the (discretized) Darboux or Laplace equations describing the lattices. An example of localized tau functions and eigenfunction potentials is given.

A family of complex potentials with real spectrum

We consider a two-parameter non hermitean quantum-mechanical hamiltonian that is invariant under the combined effects of parity and time reversal transformation. Numerical investigation shows that for some values of the potential parameters the hamiltonian operator supports real eigenvalues and localized eigenfunctions. In contrast with other PT symmetric models, which require special integration paths in the complex plane, our model is integrable along a line parallel to the real axis.

Orthonormal sets of localized functions for a Landau level

Maximal sets of commuting magnetic translations are used for constructing a set of eigenfunctions for a Landau level on a von Neumann–Gabor lattice. Localization and orthogonality turn out to be two conflicting features of this set. It is shown how to construct complete orthonormal sets of optimally localized eigenfunctions on a von Neumann–Gabor lattice for each Landau level. By using the Balian–Low theorem it is pointed out that the uncertainties of the orbit center coordinates cannot both be made finite.

Distributions of Localized Eigenvalues of Laplacians on Post Critically Finite Self-Similar Sets

In this paper, we study distributions of eigenvalues corresponding to localized eigenfunctions of Laplacians on p.c.f. self-similar sets. Precisely, we divide the eigenvalue counting functionρ(x) of a Laplacian into two parts,ρW(x) andρF(x), whereρW(x) is the counting function of localized eigenvalues andρF(x) is the counting function of non-localized (global) eigenvalues. We study asymptotic behaviors ofρW(x) andρF(x) asx→∞. It is shown thatρW(x)≈xdS/2wheredSis the spectral exponent. On the other hand, for a class of Laplacians, including the standard Laplacian on the Sierpinski gasket,ρF(x)≈xκFfor someκF<dS/2. So localized eigenfunctions dominate global eigenfunctions in such cases.

Rate of quantum ergodicity in Euclidean billiards

For a large class of quantized ergodic flows the quantum ergodicity theorem states that almost all eigenfunctions become equidistributed in the semiclassical limit. In this work we give a short introduction to the formulation of the quantum ergodicity theorem for general observables in terms of pseudodifferential operators and show that it is equivalent to the semiclassical eigenfunction hypothesis for the Wigner function in the case of ergodic systems. Of great importance is the rate by which the quantum-mechanical expectation values of an observable tend to their mean value. This is studied numerically for three Euclidean billiards (stadium, cosine, and cardioid billiard) using up to 6000 eigenfunctions. We find that in configuration space the rate of quantum ergodicity is strongly influenced by localized eigenfunctions such as bouncing-ball modes or scarred eigenfunctions. We give a detailed discussion and explanation of these effects using a simple but powerful model. For the rate of quantum ergodicity in momentum space we observe a slower decay. We also study the suitably normalized fluctuations of the expectation values around their mean and find good agreement with a Gaussian distribution.

Localized Eigenfunctions of the Laplacian on p.c.f. Self-Similar Sets

In this paper we consider the form of the eigenvalue counting function ρ for Laplacians on p.c.f. selfsimilar sets, a class of self-similar fractal spaces. It is known that on a p.c.f. self-similar set K the function ρ(x) O(xds/#) as x!¢, for some d s 0. We show that if there exist localized eigenfunctions (that is, a non-zero eigenfunction which vanishes on some open subset of the space) and K satisfies some additional conditions (‘ the lattice case ’) then ρ(x)x−ds/# does not converge as x!¢. We next establish a number of sufficient conditions for the existence of a localized eigenfunction in terms of the symmetries of the space K. In particular, we show that any nested fractal with more than two essential fixed points has localized eigenfunctions.

The Riccati method for the Helmholtz equation

The operator Riccati equation for the Dirichlet-to-Neumann map is derived from the exact operator factorization of the two-dimensional variable coefficient Helmholtz equation. Numerical schemes are developed for the operator Riccati equation and its variant using a local eigenfunction expansion. This leads to a practical computational method for acoustic wave propagation over large range distances, since the boundary value problem of the Helmholtz equation is reduced to ‘‘initial’’ value problems that are solved by marching in the range. The efficiency and accuracy of the method is demonstrated by numerical experiments including the plane-parallel range-dependent waveguide benchmark problem proposed by the Acoustical Society of America.

Exploring molecular vibrational motions with periodic orbits

Abstract The theory of periodic orbits for conservative Hamiltonian systems and the way that it is applied to analyse vibrational spectra of highly excited polyatomic molecules is reviewed. Applications for triatomic, tetratomic molecules and van der Waals clusters are presented. It is shown that the periodic orbit method can trace localized eigenfunctions above potential barriers which are associated with saddle-node bifurcations. Such states connect separate minima on the potential energy surface, and thus, are important for studying isomerization processes.

The correlation between hydrogen bond tunneling dynamics and the structure of benzoic acid dimers

We report a correlation between the rate of incoherent tunneling associated with proton transfer in hydrogen bonds and the structure of aromatic carboxylic acid dimers. The compressibility of the hydrogen bond in benzoic acid, specifically the oxygen–oxygen distance r(O⋅⋅O), has been measured as a function of hydrostatic pressure up to 3.2 kbar using neutron powder diffraction. All data were recorded at a temperature of 5 K. Using previously published pressure dependence NMR measurements, we have investigated the relationship between the dynamics in the quantum regime and r(O⋅⋅O) in the hydrogen bonds of benzoic acid. The incoherent tunneling rate increases exponentially with decreasing r(O⋅⋅O). This behavior is attributed to the increase in the tunneling matrix element as the potential wells and the localized eigenfunctions of the double minimum potential which characterize the system are brought into closer proximity. There is a quantitative agreement between this study, in which the hydrogen bonds are compressed by the application of pressure, and the behavior exhibited by two benzoic acid derivatives with different oxygen–oxygen distances at ambient pressure.

Quantum Chaos and the Limits of Semiclassical Prediction.

Spectral features and eigenfunctions of the kicked rotor are contrasted with those obtained in the semiclassical limit. For extended eigenfunctions, the difference in the spectra scales as ${\ensuremath{\Elzxh}}^{3/2}$ in the random matrix theory limit and eigenvalues can be uniquely identified. However, for dynamically localized eigenfunctions, the semiclassical spectrum does not resemble the exact spectrum though numerics indicate that statistical features are reproduced. We suggest that semiclassical periodic orbit theory, even with improved asymptotics, will only provide a statistical description of dynamical localization.

STATISTICAL PROPERTIES OF EIGENFUNCTIONS OF RANDOM QUASI 1D ONE-PARTICLE HAMILTONIANS

The article reviews recent analytical results concerning statistical properties of eigenfunctions of random Hamiltonians with broken time reversal symmetry describing a motion of a quantum particle in a thick wire of finite length L. It is demonstrated that the problem is equivalent to the study of properties of large Random Banded Matrices in the limit of large width of the band. Matrices of this class are relevant for a number of problems in Solid State physics and in the domain of Quantum Chaos. We find the analytical expressions for the distribution of the following quantities: i) the eigenfunction amplitude |ψ(r)|2 at given point of the sample; ii) spatial extent of the eigenfunction measured by the “inverse participation ratio” P=∫V dr|ψ(r)|4; iii) the quantity R=|ψ(r)ψ(r′)|2, points r and r′ belonging to the opposite ends of the sample. For a long sample the quantity –(ln R)/L characterizes the decay rate of a localized eigenfunction (Lyapunov exponent). Relation with available numerical results is discussed.