We develop a two-potential approach to the decay of a quasistationary state. The method enables us to obtain a simple algebraic formula for the decay width and the energy shift of a metastable state. The quasiclassical limit of the width found leads to the well-known Gamow formula, this time with a well-defiend preexponential factor. The energy shift together with its quasiclassical limit is obtained in closed form.

We present a high-precision experimental verification of the universality of f(α), the fractal dimension of the subset of the attractor with scaling index α, at the transition from quasiperiodicity to chaos with the golden-mean winding number. The experimental system is cooled extrinsic p-type Ge, which is voltage biased to produce a spatially dependent active oscillation.

For a molecular system with an odd number of electrons, the electronic Hamiltonian possesses a twofold Kramers degeneracy for all nuclear configurations. It is shown that this leads to a non-Abelian gauge-field term in the effective nuclear Hamiltonian, with permissible gauge transformations belonging to SU(2). This should lead to observable effects analogous to the Berry phase for the nondegenerate case. In particular, an atom in a slowly rotating electric field exhibits changes in its component of angular momentum along the field because of the non-Abelian phase factor.

We show that Nosromanē mechanics provides a link between computer simulations of nonequilibrium processes and real-world experiments. Reversible Nose¯ equations of motion, when used to constrain non- equilibrium boundary regions, generate stable dissipative behavior within an adjoining bulk sample governed by Newton’s equations of motion. Thus, irreversible behavior consistent with the second law of thermodynamics arises from completely reversible microscopic motion. Loschmidt’s reversibility paradox is surmounted by this Nose¯-Newton system, because the steady-state nonequilibrium probability density in the many-body phase space is confined to a zero-volume attractor.

A new decomposition of the multichannel scattering matrix, valid near an isolated, narrow resonance, is presented. It is consistent with the theory of single-channel resonances, and readily provides transition probabilities for inelastic resonant collisions. Some features of the formalism are illustrated by a vibration-rotation resonance in three-dimensional H + CO.

This paper demonstrates that a wide range of driven, overdamped, nonlinear systems can exhibit negative differential resistance, where the equilibrium amplitude of oscillation decreases as the amplitude of the driving force increases. As an example we examine the response of a parametrically driven bistable electronic circuit to both a dichotomous and a sinusoidal driving force. A general condition is derived for the onset of negative damping in the presence of a large-amplitude modulation.

The continuum theory of self-avoiding D-dimensional manifolds with fixed connectivity, in d dimensions, is considered. The exponent γ is argued to exist only for integer D It is shown that its ε=4D-(2-D)d expansion is not universal at 0(ε), but depends on the shape of the manifold’s boundary. γ is calculated for a hyperellipsoid. New universal contact exponents are given for various points of a crumpled manifold.

We show that a duality transformation (connecting integrable Hamiltonian systems) recently discovered by Hietarinta and co-workers in two dimensions at a classical level can be directly established in N dimensions from a quantum formulation of the problem. Further, we show duality to be valid for a much larger class of dynamical systems and to be not necessarily unique. We speculate that this nonuniqueness could be a characteristic of separable systems.

The fractal invariant measure of chaotic strange attractors can be approximated systematically by the set of unstable n-periodic orbits of increasing n. Algorithms for extracting the periodic orbits from a chaotic time series and for calculating their stabilities are presented. With this information alone, important properties like the topological entropy and the Hausdorff dimension can be calculated.

Sample rotation is shown to induce frequency splittings in nuclear-quadrupole-resonance spectra. The splittings are interpreted both as a manifestation of Berry's phase, associated with an adiabatically changing Hamiltonian, and as a result of a fictitious magnetic field, associated with a rotating-frame transformation. Real and fictitious fields are contrasted. Related effects are predicted in other magnetic resonance experiments that involve sample rotation.

A new type of disorder parameters is defined. By investigation of their properties it is concluded that the $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$ nonlinear $\sigma$ models exhibit a phase transition for $D\ge 2$; similarly for gauge theories for $D\ge 3$. For continuous spins and groups the low-temperature phase contains massless excitations.

It is argued in the context of coupled quadratic maps that macroscopically chaotic states do not occur in many-body systems with local interactions and random initial conditions. Such systems can exhibit chaos, but only locally; their collective behavior is periodic or stationary. The phase diagram for the coupled-map system as a function of control parameter and noise is presented, and the universality classes of the phase transitions identified.

Explicit scattering solutions of the one-dimensional Schrödinger equation are given. A one-parameter family of the potentials considered here causes the same scattering at all energies. The previously published explicit examples of nonuniqueness in the one-dimensional inverse quantum problem are special cases of the potentials given here.

The quantum hydrodynamics of an incompressible inviscid fluid in $R^2$ or $R^3$ is described by means of representations of the group of volume-preserving diffeomorphisms. In $R^2$ we obtain the surprising result that within the framework of the Euler equations point vortices cannot be consistently quantized, and the encouraging outcome that vortex filaments can be. We also discuss analogs of such configurations in $R^3$.

Circle maps are used to study the "spectrum of singularities" of trajectories on two-tori with golden-mean winding numbers at the onset of chaos. Within a renormalization-group approach, the effect of a deviation from criticality is understood as a crossover from a strong- to a weak-coupling fixed point.

Assuming that the chaotic time history of a single variable in a differential equation possessing a strange attractor can be represented as a random superposition of deterministic ‘‘structures,’’ we predict the power spectral density. We justify the assumption for perturbations of nonlinear Hamiltonian oscillators ans compare our predictions with computations on versions of Duffing’s equation.

Combinatorial optimization problems such as the determination of the ground-state energy of Ising spin-glasses, the construction of low-autocorrelation binary sequences, and the traveling salesman problem are analyzed with use of zero-temperature scaling concepts. The difficulty of obtaining good ground states with most heuristic algorithms is influenced by the number of local minima. It is shown that the number of such minima is related to the thermal eigenvalue at the zero-temperature fixed point.

We analyze long-range correlations in quantum spectra within the framework of periodic-orbit theory and find excellent agreement between level clustering in quantum spectra and semiclassical predictions. The analysis demonstrates how classical chaotic motion is reflected in the spectral properties of the corresponding quantum system.

A new geometric phase factor is defined for any cyclic evolution of a quantum system. This is independent of the phase factor relating the initial and final state vectors and the Hamiltonian, for a given projection of the evolution on the projective space of rays of the Hilbert space. Some applications, including the Aharonov-Bohm effect, are considered. For the special case of adiabatic evolution, this phase factor is a gauge-invariant generalization of the one found by Berry.

It was recently predicted that nondiffracting beams, with beam spots as small as a few wavelengths, can exist and propagate in free space. We report the first experimental investigation of these beams.