Using the spinor approach, we calculate exactly the complete spectrum of the transfer matrix for the finite-size, planar Ising model with adjustable boundary conditions. The mass gap is found to vanish exponentially or algebraically, depending on the boundary conditions used. These results are compared with recent numerical simulations, and with random-walk and capillary-wave arguments.

We obtain rigorous bounds on the long-time behavior of the densities ${\rho }_A\mathrm{}\left(t\right)\mathrm{}$ and ${\rho }_B\mathrm{}\left(t\right)\mathrm{}$ of species $A$ and $B$, which diffuse and annihilate upon meeting, i.e., $A+B\to \mathrm{inert}$. For equal initial densities ${\rho }_A\mathrm{}\left(0\right)={\rho }_B\mathrm{}\left(0\right)\mathrm{}$, the density goes to zero asymptotically as $t^{\frac{-d}{4}}$ for dimensions $d\le 4$ and as $t^{-1\mathrm{}}$ for $d\ge 4$. When ${\rho }_A\mathrm{}\left(0\right)<{\rho }_B\mathrm{}\left(0\right)\mathrm{}$, ${\rho }_A\mathrm{}\left(t\right)\mathrm{}\sim \exp \left({-\lambda }_1\sqrt{t}\right)$, ${\rho }_A\mathrm{}\left(t\right)\mathrm{}\sim \exp \left(\frac{{-\lambda }_{2}t}{\mathrm{In}t}\right)$ for $d=1,2, \mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{y},\mathrm{a}\mathrm{n}\mathrm{d}{\rho }_A\mathrm{}\left(t\right)\mathrm{}\sim \exp \left({-\lambda }_{d}t\right)$ for $d\ge 3$.

A particle moving in two dimensions in a repulsive $r^{-2\mathrm{}}$ potential, and in the presence of a magnetic flux line, is examined as an example of a system for which the Schrödinger Hamiltonian is not essentially self-adjoint. The path-integral propagator is, nevertheless, shown to exist, and to define a unique self-adjoint extension of the Hamiltonian, corresponding to specific boundary conditions at the origin. Physical implications are discussed.

An analytical solution for one-dimensional diffusion in a Gaussian random potential is presented. In the long-time limit, the logarithm of the average population at the center, $\mathrm{In}\left(〈P〉\right)$, grows as fast as $t^{\frac{3}{2}}$, disproving some former estimates that $\mathrm{In}\left(〈P〉\right)$ increases at the rate of $t^2$. Numerical simulations have confirmed the theoretical solution.

We present the wavelet transform as a mathematical microscope which is well suited for studying the local scaling properties of fractal objects. We apply this technique to probability measures on self-similar Cantor sets and to the golden-mean trajectories on two-tori at the onset of chaos.

In analogy to the proposed gravitational "fifth" force, limits can be set on the range $\lambda$ and the strength $\beta$ of an electrical fifth force. Specifically we consider the potential $V=\frac{(e^2}{r)}(1+{\beta e}^{-r\lambda })$ and show that there is a region near $\lambda =1\mathrm{}$ μm where existing experiments set relatively poor limits on $\beta$.

The classical stadium billiard, which is known to be ergodic and strongly mixing, is shown to have strongly localized quantum eigenstates which persist up to infinite energy, or alternately which survive the $\hslash \to 0$ limit. Consistent with the theorems of Schnirlman, Zelditch, and Colin de Verdiere, the states which we show are highly localized are collectively of zero measure (though there are infinitely many) as $E\to \infty$ or as $\hslash \to 0$. However, more relevant to the experimental world is the fact that such localized states make up a finite and calculable fraction of the quantum eigenstates at finite energy.

The Bohr-Sommerfeld and Maslov phases, which occur in the Einstein-Brillouin-Keller quantization rule of semiclassical mechanics, are shown to be examples of Berry's phase. The generalization of Berry's phase due to Aharonov and Anandan, which does not require adiabaticity, is employed.

The first experimental observation of a bifurcation sequence of patterns in a reaction-diffusion system is reported. The experiment on a one-dimensional system, maintained away from equilibrium, reveals steady, periodic, quasiperiodic, frequency-locked, period-doubled, and chaotic spatiotemporal states. Results from a model reaction-diffusion system agree qualitatively with the experiment and provide insight into the physical mechanism that drives the observed behavior.

In this Letter, we describe the connection between the classical concepts of nonintegrability and chaos and the quantum concept of dynamical symmetry breaking. The existence of unbroken dynamical symmetry implies integrability of the mean-field motion in a quantum phase space, defined as a symplectic coherent-state parametrization of the coset space of the overall dynamical group. Broken dynamical symmetry leads to nonintegrability, and thus chaotic solutions to Hamilton's equations in the quantum phase space. We illustrate the general ideas with results obtained for a model of two coupled spins.

Highly excited hydrogen atoms have been exposed to an intense pulse of microwaves with frequency near the initial classical electron orbit frequency. Measurements were made of the probability distribution for different bound states after the pulse. Structure in such distributions is found to be near classical, and arises from the presence of a classical resonance island in action-angle space. The measured distributions also yield values of electron localization length in action space that are in agreement with quantum localization theory and with numerical values based on quantum mechanics.

We report the first measurements of microwave excitation and ionization of excited hydrogen atoms for scaled frequencies $n_0^3\omega$ up to 2.8. Classical 3D calculations which directly model this 36.021-GHz experiment agree quite well for $n_0^3\omega <1\mathrm{}$, agree less well for $1\le n_0^3\omega \le 2$, and do not agree for $n_0^3\omega >2\mathrm{}$. This supports theoretical predictions that as $n_0^3\omega$ rises above 1, quantal ionization threshold fields rise above those for the onset of classical chaos; however, the data continue to reveal local stability near certain rational frequency ratios that recalls classical behavior.

A two-dimensional condensed-matter lattice model is presented which exhibits a nonzero quantization of the Hall conductance ${\sigma }^{\mathrm{xy}}$ in the absence of an external magnetic field. Massless fermions without spectral doubling occur at critical values of the model parameters, and exhibit the so-called "parity anomaly" of (2+1)-dimensional field theories.

If the magnetic flux trapped in a superconducting quantum-interference device has observable values $\pm \frac{h}{2e}$, which alternate in time, and if the value at a given moment is unknown, it is in general impossible to determine that value without affecting the results of subsequent observations.

We present new results of Monte Carlo simulations for self-avoiding walks on randomly diluted square and simple-cubic lattices performed very close to the percolation thresholds. Our results indicate the asymptotic behavior of the walk dimension to be rather similar to the undiluted lattice even at critical dilution.

A general cluster updating algorithm for Monte Carlo simulations is presented. The method contains arbitrary probability functions which can be used to minimize the relaxation time. It is applicable to systems where the interaction energy has a global (discrete or continuous) symmetry. Special cases cover standard site-by-site updating as well as the cluster updating method proposed by Swendsen and Wang for Ising and Potts models.

Using stored spin-polarized ultracold neutrons subjected to a temporally varying magnetic field, we have demonstrated a manifestation of Berry's phase. We verified several properties of Berry's phase which include its solid-angle dependency and additivity for multiple excursions along the same path in parameter space.

Experiments are presented which demonstrate the conversation of certain integrals of motion during Hamiltonian chaos where the conservation of some of the integrals has been broken by the overlap of resonances. Experimentally, conservation laws are demonstrated through the observation of restrictions on transport, in phase space, of chaotic trajectories. Transport of plasma ions during stochastic motion in an electrostatic wave is followed by means of laser-induced fluorescence on metastable "test particles" produced by optical pumping.

We generalize the concept of level spacings to dissipative quantum maps. For periodically kicked tops with damping, we find linear and cubic level repulsions under conditions of classically regular and chaotic motion, respectively. The numerically obtained spacing distribution for the chaotic top appears to be universal: It compares favorably with the spacing distribution of general complex matrices of large dimension, the analytical form of which we also present.

We show that the usual derivations of the Lorentz-Dirac equation are not valid at points where the charge's path is a nonanalytic function of time, and that such points inevitably appear when the applied force is freely alterable in the future, i.e, when one considers questions of causality. This enables us to avoid causality violation in classical radiation reaction, making the usual hedge of quantum effects at small distances quite unnecessary.