Abstract: We define a hopping Hamiltonian for independent electrons on a two-dimensional quasiperiodic Penrose lattice. This problem is then investigated numerically, up to systems of 3126 sites, and for various boundary conditions. We find the following results for the density of states: (1) There is a central peak of zero width at zero energy, consisting of about 10% of the total number of states. (2) These states are strictly localized; we calculate the wave functions explicitly. (3) The remainder of the states lie in two bands, symmetric about zero energy, separated from the localized states by a finite gap ${E}_{0}$.

Abstract: We describe the integrable structure of solutions of the nonlinear Schrodinger (NLS) equation under periodic and quasiperiodic boundary conditions. We focus on those aspects of the exact theory which reveal the behavior of these solutions under perturbations of initial conditions (i.e. linearized instabilities), and the effects of slow modulations in space and time, perhaps in the presence of external perturbations. These results and methods continue the investigations of Ercolani, Flaschka, Forest and McLaughlin [1–7] on Korteweg-deVries (KdV), sine-Gordon (sG) and sinh-Gordon wavetrains. Our purpose here is to document the corresponding features of NLS solutions; the rigorous analysis that underlies this paper derives from [1–7] and will appear in the thesis of Lee [8].

Abstract: We define a one-parameter family of hopping Hamiltonians with an on-site potential, for independent electrons on a two-dimensional quasiperiodic Penrose lattice. The resulting models include the vibrational modes of the lattice. This problem is then investigated numerically\char22{}exploiting the symmetries of the model including scale invariance, up to systems of 3126 sites\char22{}and for various boundary conditions. We find the following results for the density of states. (1) There is a peak of zero width at a known energy. (2) We calculate the fraction of states in this central peak, assuming point (5) below. (3) These states are strictly localized; we calculate the wave functions explicitly. (4) The remainder of the states lie in various bands separated by band gaps; this band structure is discussed in the limit of a large on-site potential. (5) There is strong numerical evidence that the energy eigenvalues do not cross, as the on-site potential strength is varied, in the thermodynamic limit.

Abstract: This lecture deals with the theory of quasiperiodic tilings, and more generally with quasiperiodic patterns. We present the ideas introduced in [9] and developed in [10].

Abstract: I define a hopping Hamiltonian for independent electrons on a two-dimensional, infinite, quasiperiodic Penrose lattice with a particular on-site potential, depending upon a parameter r. I then find the exact ground-state wave function for this Hamiltonian. The wave function is shown numerically to have a power-law decay from the origin, and the exponent is determined numerically. The wave function may or may not be normalizable over the infinite lattice, depending upon the parameter r. The wave function is then demonstrated to be self-similar, in that the wave function for two identical regions of the lattice is the same, except for a scale factor. The scaling of the wave function is discussed, and a bound for the decay of the wave function established. Finally, I determine exactly the scaling distribution for the wave function, and thus calculate exactly the previously introduced decay exponent. The wave function is shown to have a critical value of the parameter r, above which the wave function is normalizable and thus localized, while below the wave function has a power-law decay but is not normalizable.

Abstract: We study the plasma excitations of an array of two-dimensional electron-gas layers arranged in a Fibonacci sequence. The plasmon spectrum is analyzed by a transfer-matrix method. It is shown that the spectrum is always "critical," i.e., it corresponds to neither extended nor localized plasmon states. This property is reflected in the calculated Raman intensities.

Abstract: We have magnetron-sputtered a series of Mo-V superlattices which have a quasiperiodic layering in the growth direction. We have used the Fibonacci series as the generating rule for the nearly periodic structures and have verified their structure using high-angle x-ray diffractometry. The superconducting transition temperatures slowly increase as a function of the quasiperiodic wavelength ${\ensuremath{\Lambda}}_{F}$, while the initial upper-critical-field slopes parallel to the films decrease with increasing wavelength, and the parallel upper-critical-field curves ${H}_{c2\ensuremath{\parallel}}$ display a two-dimensional behavior that is not consistent with current ideas about critical-field behavior in multilayers.

Abstract: Periodic instabilities of frequency f/sub 0/ in an electron-hole plasma in Ge are excited with an external perturbation V/sub 1/ sin(2..pi..f/sub 1/t). Varying V/sub 1/ and f/sub 1/ results in quasiperiodic, mode-locked, and chaotic states comparable to those expected theoretically. Increasing V/sub 1/ while keeping the ratio f/sub 0//f/sub 1/ fixed at the golden mean (1.6180 . . .), we observe a universal transition to chaos, in quantitative agreement with theory. Driving the instability simultaneously at two incommensurate frequencies, f/sub 1/ and f/sub 2/, we observe transitions between three-frequency quasiperiodicity, two-frequency mode locking, and chaos.

Abstract: We study and compare plasmon spectra in semiconductor superlattices with periodic, quasiperiodic (e.g., Fibonacci sequence), and random spacings. Novel mode structures are found in the quasiperiodic cases, particularly for low values of level broadening that arises from inherent disorder effects. We discuss critically the observability of this novel mode structure via light scattering studies in experimentally realizable superlattices. In addition, our results suggest that the finite Fibonacci-sequence superlattice may be considered as a partially ordered-layer lattice with no long-range positional order, but with strong short-range order, since our calculated plasmon spectra exhibit very little difference between the finite Fibonacci sequence and the periodic abaab superlattice.

Abstract: The structure of one-dimensional quasicrystal is reconsidered so that we can intuitively understand its electronic properties. It is shown that the Fibonacci sequence is obtained from a regular crystal by successive application of periodic modulations, which have the periods increasing as n , n 2 , n 3 , ··· ( n = 3) and are self-similar. Explanations are given for the peculiar spectra of electronic states on a one-dimensional quasiperiodic lattice as a result of the modulations applied to the crystal. Differences are also briefly discussed for the phonon and electronic band structure.

Abstract: A model of two mutally incommensurate, interacting, dynamical, many-body systems is presented and solved. Quasiperiodic forms are shown to describe both the dc properties and the complete set of linear excitations. By use of one system to represent the crystal lattice and the other a charge-density wave (CDW), all the recently discovered, and as yet unexplained, electromechanical properties of CDW conductors are shown to occur in this model. The internal degrees of freedom of the CDW are shown to be of central importance.

Abstract: We predict anomalous x-ray transmission through a quasicrystal slab in the Laue geometry using two models: a Fibonacci quasiperiodic lattice and a random packing of icosahedra. We used the two-beam approximation in the dynamical theory of x-ray diffraction. Multiple-beam effects are also discussed.

Abstract: The dynamics of energy transfer is discussed for a model system in which two ligands are separated by a heavy atom. Numerical and analytical results are given for the case that each ligand is a CC. In the quasiperiodic regime, the dynamics are interpreted using perturbation theory. Local group modes involved in an intramolecular energy localization which can occur in this regime are identified. An approximate separation of the primarily ligand–ligand motions from the primarily ligand–metal–ligand ones occurs in the clearly quasiperiodic regime and also at an energy where the power spectra of the bond coordinates are "grassy." The overall analysis is used to make predictions for systems with larger ligands, when the primarily metal atom–ligand modes are, as above, approximately separable from the primarily intraligand ones.

Abstract: The 27 Al and the 55 Mn nuclear magnetic resonance powder pattern lineshapes have been obtained in icosahedral and decagonal ( T phase) Al-Mn quasiperiodic crystals, and are compared to that of orthorhombic Al 6 Mn. The quasiperiodic crystals yield much broader spectra with little resolved structure. The quadrupole and Knight shift parameters for the 55 Mn resonance in orthorhombic Al 6 Mn have been determined as | v Q | = 0.76 MHz,Kax = −2.7 × 10 −4 , K iso = + 5 × 10 −3 . The results imply that Al 6 Mn and the quasiperiodic crystals have similar electronic and magnetic properties.

Abstract: It is a major unsolved problem to understand the cause of spatial symmetry (specifically crystalline symmetry) in low-temperature bulk matter(1 5) The essence of the problem--why energy ground states tend to be periodic--has been essentially solved for one-dimensional models, (6'71 but there are what could be interpreted as "counterexamples" in two and three dimensions, (8-1~ ie, simple models which have quasiperiodic, but no periodic, ground states Periodic configurations are special in many ways They exhibit long range order in that the configuration in one region determines the configuration even in very distant regions Also, they are homogeneous in the sense that the configuration is essentially identical in any two congruent regions much larger than the period We will show that energy ground states tend to have this homogeneity property, at least in an average sense This seems to be the first general, qualitative result on the spatial symmetry of ground states for dimensions higher than one

Abstract: The work of Hill (1985a) on the low-degree 5 min eigenfrequency spectrum of the Sun based on differential radius observations combined with Doppler shift and total irradiance observations has been extended to include the work of Harvey and Duvall (1984) and Libbrecht and Zirin (1986). The differences between eigenfrequencies obtained in this analysis are compared to the predictions of asymptotic theory, and the deviations between observation and theory are observed to be ≈4 times larger than expected based on estimated accuracy of eigenfrequency determinations. These deviations are tested for departures from predictions of asymptotic theory which are quasi-periodic as a function of radial ordern and degreel. It is observed that the superposition of the seven distributions of frequency differences vsn obtained forl=0, ..., 6 generates an overdetermined quasi-periodic function when Δn/Δl=0.420±0.018 and Δv/Δl=−0.35 μHz in the superposition process. The probability that this quasi-periodic parent function is obtained from seven independent random distributions is estimated to be ≈1.2×10−7. Numerical experiments performed with theoretical eigenfrequency spectra demonstrate that the existence of a quasiperiodic behaviour in the superposed spectrum of frequency differences is physically plausible and that the parameters used in the superposition process are consistent with theory. One significant theoretical quasiperiodic behaviour is obtained for Δn/Δl=0.399. By comparing the properties of the observed quasi-periodic behaviour with those obtained in the numerical experiments, we infer that the location of the region which leads to the greatest departure from asymptotic theory predictions is 0.757±0.002 solar radii, which suggests that this region is connected with the transition zone between the radiative interior and the convection zone.

Abstract: The Boussinesq equations for the Rayleigh-Benard problem have been solved by analytical and numerical methods. Two different sequences of Hopf bifurcations, leading from, stationary two-dimensional rolls to non-periodic motion, have been identified. For one of the sequences the first bifurcation results in transverse oscillations of the rolls. The next bifurcation gives quasiperiodic flow, and the sequence ends in chaotic motion after the third instability. The second route is characterized by waves in the periodic regime, travelling along the rolls. Thereafter two quasiperiodic regimes follow with two and three frequencies, respectively. Both types of sequences have been detected in the experiments reported by Gollub and Benson (1980). The regime of travelling waves is also analysed by a perturbation method.

Abstract: High quality experimental data have been taken on a convection cell containing a dilute 3He−4He solution. We discuss some problems with the determination of dimension and entropy for experimental data, and compare the results to detailed Poincare sections. At the chaotic transition, we show the behavior of dimension and entropy as a function of Rayleigh number.

Abstract: A renormalisation procedure is constructed for the one-dimensional Schrodinger equation with a quasiperiodic potential. The renormalisation transformation has a trivial fixed point describing the behaviour of the system near the unperturbed free motion, and a non-trivial fixed point corresponding to the critical case. A number system of an irrational base is introduced for the scaling of the spectra.

Abstract: The author studies the diffraction spectrum and the structure factor of quasicrystalline, aperiodic, linear arrays with two arbitrary characteristic measure lengths, produced by a fairly general family of generating rules depending on two parameters. The calculation is performed by actually constructing the two-dimensional periodic structure whose projection will result in the desired linear aperiodic array. Distributions of some sequences of irrational numbers modulo 1 are derived as a byproduct of the main subject.

Abstract: The problem of a particle inside a rigid box with one of the walls oscillating periodically in time is studied quantum mechanically In the classical limit, this model was introduced by Fermi in the context of cosmic ray physics The classical solutions can go from being quasiperiodic to chaotic, as a function of the amplitude of the wall oscillation In the quantum case, we calculate the spectral properties of the corresponding evolution operator, ie: the quasi-energy eigenvalues and eigenvectors The specific form of the wall oscillation, eg \(\ell (t) = \sqrt {1 + 2\delta \left| t \right|}\), with |t| ≤ 1/2, and l(t + 1) = l(t) , is essential to the solutions presented here It is found that as ℏ increases with δfixed, the nearest neighbor separation between quasi-energy eigenvalues changes from showing no energy level repulsion to energy level repulsion This transition, from Poisson-like statistics to Gaussian-Orthogonal-Ensemble-like statistics is tested by looking at the distribution of quasi-energy level nearest neighboor separations and the Δ3(L) statistics These results are also correlated to a transition between localized to extended states in energy space The possible relevance of the results presented here to experiments in quasi-one-dimensional atoms is also discussed