We study confinement effects on the energy eigenvalues, dipole moments and Einstein coefficients of a model harmonic oscillator restricted by two impenetrable walls placed either symmetrically or asymmetrically with respect to the potential minimum. The calculations are made using perturbation theory as a function of the position of the potential minimum with respect to the bounding walls. For small boxes, the energy levels resemble more closely those of a free particle in a box, than those of an unbounded harmonic oscillator. When the size of the box increases, the lowest energy levels become more similar to those of the unbounded harmonic oscillator, but the highest energy levels remain similar to those of the free particle in a box. We also show that the selection rules for the confined harmonic oscillator are not the same as those of the unbounded harmonic oscillator.

We calculate for the first time the distribution of energy widths for Anderson localized states at a surface and show that electrical noise is produced with a 1//spectrum. The distribution has a natural cutoff at widths of h/lc, where lc<^txp(S'^L/2) and S'^ measures the disorder; 4 is typically many hours. The scanning tunneling microscope is proposed as the ideal device for observing these fluctuations.PACS numbers: 73.20.-r, 71.55.Jv Low-energy electron diffraction from surfaces is well understood for ordered systems, ^*^ and even for certain special kinds of weak disorder some progress has been made in describing diffuse scattering,^ For very strongly disordered surfaces no theory at all exists and the radical effects of such strong disorder present a situation where conventional spectroscopies are inadequate for probing these phenomena: New experiments are required. The radical effect of strong disorder on our theoretical picture is to induce Anderson localization^"*^ in the electron wave functions so that they no longer spread over the whole system. This localization in turn implies that the energy levels of the system are very narrow as opposed to the ordered system where the states form a continuum. The widths d of these localized states are much too narrow to be observed by conventional spectroscopy whose resolution can rarely exceed 1 meV.^ So small is d in general that h/d is of the order of seconds.The main message of this Letter is that we have developed a formalism that enables us to calculate the statistical distribution of level widths. Analytic solutions show the following: (i) Most of the levels have a \/d distribution of widths, (ii) This distribution implies 1// noise in the total surface charge, (iii) Saturation of the widths occurs at very small values of d with h/d being many hours or even days, (iv) The noise*^'*"* arising in surface charge and its deviation from a 1// law can be used as a spectroscopy of the level widths, (v) The cutoff is related to the diffusion-limited frequency response of the disordered material in transmission, (vi) The cutoff has implications for the approach to thermodynamic electronic equilibrium in disordered systems which may take many days and invites comparison with spin-glasses. (vii) Scanning tunneling microscopy*^ has the potential to examine very small areas and explore the cutoff in the noise spectrum.We treat the mathematical side of the problem by solving a simplified model of the surface exactly using the technique of the symmetric group.*^ Our model approximates the highly disordered surface by an array of one-dimensional chains of atoms, the chains running in three directions mutually at right angles. The density of atoms we take to be the same as in the real surface. This model can only describe the limit of strong disorder because the chains exhibit only localized behavior, but in this limit we believe that the model reproduces the essential features of a localized system. However, inelastic scattering due to the phonons is not included and our co...

We show that the key transport states, insulating and conducting, of large regular networks of scatterers can be described generically by negative and zero Lyapunov exponents, respectively, of Möbius maps that relate the scattering matrix of systems with successive sizes. The conductive phase is represented by weakly chaotic attractors that have been linked with anomalous transport and ergodicity breaking. Our conclusions, verified for serial as well as parallel stub and ring structures, reveal that mesoscopic behavior results from a drastic reduction of degrees of freedom.

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We study the effect of left-right symmetry on the distribution of the parametric derivative of the dimensionless conductance T with respect to an external parameter X , partial differentialT/ partial differentialX , of ballistic chaotic cavities with two leads, each supporting N propagating modes. We show that T and partial differentialT/ partial differentialX are linearly uncorrelated for any N . For N=1 we calculate the distribution of partial differentialT/ partial differentialX in the presence and absence of time-reversal invariance. In both cases, it has a logarithmic singularity at zero derivative and algebraic tails with an exponent different from the one of the asymmetric case. We also obtain explicit analytical results for the mean and variance of the distribution of partial differentialT/ partial differentialX for arbitrary N . Numerical simulations are performed for N=5 and 10 to show that the distribution P ( partial differentialT/ partial differentialX) tends towards a Gaussian one when N increases.