The change of the crossing parameter (essentially the phase) between sequential slow separatrix crossings is calculated for Hamiltonian systems with one degree of freedom. Combined with the previous separatrix crossing analysis, these results reduce the dynamics of adiabatic systems with separatrices to a map. This map determines whether a trajectory leaving a given separatrix lobe is ultimately captured by the other lobe. Averaging these results over initial phase yields the reaction probability, which does not asymptote to the fully phase-mixed result even for arbitrarily long times between separatrix crossings.

It is shown that chaotic trajectories in volume-preserving flows, ṙ${=\mathrm{u}}_{\mathrm{\epsilon }}$(x,y,z,t), which are arbitrarily close to integrability, 0<ε≪1, can be either trapped or diffusive throughout the available space. A classification of these flows is proposed which both distinguishes and predicts the appropriate type of behavior. In the unbounded case, a new mechanism of diffusion is found which combines motion on the resonances with an adiabatic drift. This process is reminiscent of Arnold diffusion.

The periodic-orbit quantization approach of Gutzwiller, which is a semiclassical approximation of Feynman’s path-integral formalism, is used to calculate finite-resolution densities of states for a Hamiltonian system whose classical motion is dominated by chaos, viz., the hydrogen atom in a uniform magnetic field. In spectral regions where the resolution obtained is larger than the mean level spacing, it is possible to extract approximate eigenvalues of the quantized system. The particular role of stable periodic orbits and their direct quantization are elaborated. Remarkable agreement with exact quantum calculations results.

We show the existence of a generic integrate-and-fire phenomenon which separates the transition to complete phase locking from the transition to chaos. We argue that our picture provides a natural explanation of "missing" hysteretic bands in the Belousov-Zhabotinskii reaction. In addition, we explain the origin of experimentally observed "deviations from universality" and bumps in the Arnold tongues in driven relaxation oscillators.

The thermally activated escape from a metastable well is studied in a region where both recrossing and depletion effects are important. It is shown that the relevant variable for the escape dynamics is the energy in the unstable normal mode at the barrier. In terms of this quantity the theory becomes easily perceptible and allows for a calculation of the escape rate in the turnover region between the low damping and strong damping Kramers limits.

It is shown that the geometric phase accompanying an arbitrary cyclic change of the state vector can be naturally understood as a canonical phase term in the coherent-state path integral. The adiabatic phase is shown to be derived as a part of the canonical phase.

A quantum system whose classical limit is at the critical threshold to global chaos is studied, i.e., in the small Planck's constant limit the last invariant torus is about to disappear. The quantum states are compared with classical phase-space structure near this torus. We exhibit the extension of the scaling properties of the latter to the quantum regime and find a new scaling of the quantum eigenvalues.

The Dirac equation for hydrogenic atoms has a well known instability when $z\alpha >1\mathrm{}$. A similar instability occurs for the "relativistic Schrödinger equation" with $\frac{p^2}{2m}$ replaced by ${(p^2{c}^2+m^2{c}^4)}^{\frac{1}{2}}-m{c}^2$ at $z\alpha =\frac{2}{\pi }$. These instabilities concern only the product $z\alpha$, but when the many-electron-many-nucleus problem is examined (in the relativistic Schrödinger theory) we find that a bound on $\alpha$ alone (independent of $z$) is then required for stability. If $\alpha <\frac{1}{94}$ we find that stability occurs all the way up to the critical value $z\alpha =\frac{2}{\pi }$, whereas if $\alpha >\frac{128}{15\pi }$ then the system is unstable for all values of $z$. Some implications of these findings are also discussed.

The eigenenergies for the spin $S$ generalization of the critical $\mathrm{XXZ}$ model are calculated by solving numerically, for finite chains, their associated Bethe-Ansatz equations. The conformal anomaly and scaling dimensions are obtained from the conformal invariance of the infinite chain. Our results reveal that these models are governed by a conformal field theory with central charge $c=\frac{3S}{\mathrm{}(S+1\mathrm{})\mathrm{}}$ and having continuously varying critical exponents, like the eight-vertex model. We also show that they are related to a large class of unitary conformal field theories.

The problem of instanton interactions is considered for the case of dissipative tunneling out of a metastable state. The complex ground-state energy is expressed in terms of the pressure of the interacting instanton gas evaluated at a purely imaginary fugacity. With the Yang-Lee theory it is shown that for sufficiently small barrier height a gas-liquid condensation occurs. As a consequence the decay rate becomes much smaller than the corresponding ideal gas result. This is compared with experiments on tunneling in strongly damped superconducting quantum-interference devices.

We consider polymers (self-avoiding walks) on a randomly triangulated planar surface. The partition function of $L$ polymer lines tied by their extremities, in a fluctuating metric, is calculated exactly. The exact infinite conformal spectra so derived are in complete agreement with the exponents found recently for 2D quantum gravity by Knizhnik, Polyakov, and Zamolodchikov.

We show that strong spatial correlations in a random resistor network can dramatically alter its transport properties. We calculate the average logarithmic resistance of a topologically one-dimensional model characterized by a random multiplicative process. We find a transport exponent that depends explicitly on the form of the spatial correlations; we also find that this problem is related to diffusion in the presence of correlated random fields.

The phenomenological relations between thermodynamic fluxes and forces are normally assumed to be invariant with respect to arbitrary motion of the frame of reference. We describe a breakdown of this invariance strong enough to be observable. It is shown that the diffusivity in a rotating fluid is anisotropic and also smaller in magnitude than in a fluid at rest in an inertial frame, giving rise to a diffusion analog of the Hall effect. For large Brownian particles (e.g., biological macromolecules) the diffusivity may decrease by 50% at the rotation speeds achievable in ultracentrifuges.

We discuss topological invariants for Fermi systems that have time-reversal invariance. The TK${\mathrm{N}}^2$ integers (first Chern numbers) are replaced by second Chern numbers, and Berry's phase becomes a unit quaternion, or equivalently an element of SU(2). The canonical example playing much the same role as spin ½ in a magnetic field is spin ½ in a quadrupole electric field. In particular, the associated bundles are nontrivial and have ± 1 second Chern number. The connection that governs the adiabatic evolution coincides with the symmetric SU(2) Yang-Mills instanton.

We discuss the problem of solving large linear systems of equations that arise in lattice systems with disorder. Three examples of this kind of problem are (i) computing currents in a random-resistor network, (ii) computing the fermion (quark) propagator in lattice quantum chromodynamics, and (iii) the discrete Schrödinger operator with a random potential (the Anderson model of localization). We show that the algebraic multigrid is a very effective way to compute currents in a random-resistor network. It is likely that similar techniques will apply to the other problems.

We consider a simple quantum model with a surface and prove the existence of a surface density of states. We show that the energy spectrum of the model is the union of the support of the bulk densities of states of the media forming the surface and the support of the surface density of states.

The diffusion of a Brownian particle is investigated for a class of potentials (soft bistable) which resist conventional techniques. A new method of solution based on supersymmetric quantum mechanics is shown to provide, with minimal effort, an accurate determination of the small eigenvalue of the relevant Fokker-Planck equation. The reciprocal of the small eigenvalue compares very closely with a refined definition of mean first-passage time.

General, separate formulas are derived for the quantum and the statistical fluctuations of any operator. The formalism is applied to the study of these fluctuations for the particle-number as well as the angular-momentum operators in a realistic case.

We present the first experimental results of quenching (ionization) of $\mathrm{H}(n_0=43\mathrm{}-49)$ atoms by a linearly polarized microwave electric field consisting of two frequencies (7.58 and 11.89 GHz). Adding a second frequency increases the quenching probability over that for only one frequency, especially where there is local stability at scaled frequencies near certain rational fractions, e.g., $n_0^3\omega =\frac{1}{6} \mathrm{and} \frac{1}{5}$. Some, but not all, two-frequency quench curves display a rich structure that appears to be related to that observed in some single-frequency quench curves.

A hyperinflation rule is found for one-dimensional systems with different periodicity or quasiperiodicity determined by a parameter $\alpha$. Physical systems described by transfer matrices are analyzed and the hyperinflation rule is shown to give rise to self-similarity in the space spanned by parameters of the system and $\alpha$.