The ground state energy of Sdhrödinger operators

We studye()=inf spec(-+V) and examine whene()<0 for all 0. We prove thatc ²e()–d ² for suitableV and all small ||.Research partially funded under NSF grant number DMS-9101716.     

Zero measure spectrum for the almost Mathieu operator

We study the almost Mathieu operator: (H _{, ,} u)(n)=u(n+1)+u(n-1)+ cos (2n+)u(n), onl ² (Z), and show that for all ,, and (Lebesgue) a.e. , the Lebesgue measure of its spectrum is precisely |4–2|. In particular, for ||=2 the spectrum is a zero measure cantor set. Moreover, for a large set of irrational 's (and ||=2) we show that the Hausdorff dimension of the spectrum is smaller than or equal to 1/2.Work partially supported by the GIF     

A solvable model exhibiting a glass-like transition

A nonlinear equation of motion of an overdamped oscillator exhibiting a glass-like transition at a critical coupling constant _c is presented and solved exactly. Below _c , in the fluid phase, the oscillator coordinatex(t) decays to zero, while above _c , in the amorphous phase, it decays to a nonzero infinite time limit. Near _c the motion is slowed down by a nonlinear feedback mechanism andx(t) decays exponentially to its long time limit with a relaxation time diverging as (1 – / _c )^–3/2 and (/ _c –1)^–1 for < _c and > _c respectively. At _c x(t) exhibits a power law decay proportional tot ^– with exponent -1/2.     

Geometry and combinatorics of Julia sets of real quadratic maps

For real a correspondence is made between the Julia setB forz(z–)², in the hyperbolic case, and the set of-chains±(±(±..., with the aid of Cremer's theorem. It is shown how a number of features ofB can be understood in terms of-chains. The structure ofB is determined by certain equivalence classes of-chains, fixed by orders of visitation of certain real cycles; and the bifurcation history of a given cycle can be conveniently computed via the combinatorics of-chains. The functional equations obeyed by attractive cycles are investigated, and their relation to-chains is given. The first cascade of period-doubling bifurcations is described from the point of view of the associated Julia sets and-chains. Certain Julia sets associated with the Feigenbaum function and some theorems of Lanford are discussed.Supported by NSF grant No. MCS-8104862.Supported by NSF grant No. MCS-8203325.     

Tunable UV radiation at short wavelengths (188–240 nm) generated by sum-frequency mixing in lithium borate

Sum-frequency mixing (₃=₁+₂) of UV laser radiation (₁=266 nm and 213 nm) and tunable coherent infrared light (₂=1.2–2.6 m) in lithium borate (LBO) generates radiation at short wavelengths (₃=188–242 nm). The UV radiation at ₁ is produced by the fourth and fifth harmonic of a pulsed Nd-YAG laser. The infrared light is generated with an optical parametric oscillator of beta barium borate. The phase-matching angle is measured as function of ₃ and compared with calculated values. For UV laser radiation at shorter wavelengths (173 nm₁213 nm) the calculations predict an extension of the tuning range of the sum-frequency generated at ₃ to wavelengths as short as the LBO transmission cutoff at 160 nm.     

The central limit theorem for the spectrum of the random one-dimensional Schrödinger operator

Let H_L = –d²/dt²+q(t,) be an one-dimensional random Schrödinger operator in ²(–L, L) with the classical boundary conditions. The random potential q(t,) has a form q(t, )=F(x_t), where x_t is a Brownian motion on the Euclidean v-dimensional torus, FS^v R¹ is a smooth function with the nondegenerated critical points, min_s ^v F = 0. Let are the eigenvalues of H_L) be a spectral distribution function in the volume [– L,L] and N() = lim_L(1/2L)N_L() be a corresponding limit distribution function.Theorem 1. If L then the normalized difference N _L ^* ()=[N_L() -2L·N()]2L tends (in the sense of Levi-Prokhorov) to the limit Gaussian process N^*(); N^*()0, 0, and N^*() has nondegenerated finitedimensional distributions on the spectrum (i.e., > 0). Theorem 2. The limit process N^*() is a continuous process with the locally independent increments.     

The asymmetric contact process on a finite set

The contact process onZ has one phase transition; let _c be the critical value at which the transition occurs. Let _N be the extinction time of the contact process on {0,...,N}. Durrett and Liu (1988), Durrett and Schonmann (1988), and Durrett, Schonmann, and Tanaka (1989) have respectively proved that the subcritical, supercritical, and critical phases can be characterized using a large finite system (instead ofZ) in the following way. There are constants ₁() and ₂() such that if < _c , lim _{N N} /logN = 1/₁(); if > _c , lim _N log _N /N = ₂(); if = _c , lim _{N N} /N= and lim _{N N} /N ⁴=0 in probability. In this paper we consider the asymmetric contact process onZ when it has two distinct critical values _c1< _c2. The arguments of Durrett and Liu and of Durrett and Schonmann hold for < _c1 and > _c2. We show that for [ _c1< _c2), lim _{N N} /N=-1/, (where _i is an edge speed) and for = _c2, lim _N log _N /logN=2 in probability.     

Fractional noise

Fractional noiseN(t),t 0, is a stochastic process for every , and is defined as the fractional derivative or fractional integral of white noise. For = 1 we recover Brownian motion and for = 1/2 we findf ^–1-noise. For 1/2 1, a superposition of fractional noise is related to the fractional diffusion equation.     

Nature of 2-Aminoxanthone Fluorescence

Absorption and fluorescence spectra of 2-aminoxanthone in solutions of different types at 77–350 K were studied. The existence of three bands (_{1 max} = 417 nm, ₁ = 14 ns; _{2 max} = 528 nm, ₂ = 13 ns; and _{3 max} = 565 nm, ₃ = 6 ns) in fluorescence of 2-aminoxanthone solutions has been established. It was shown that the first short-wave band was determined by deactivation of singlet excitation of an aminoxanthone molecule. The band with _{3 max} = 565 nm (depending on the concentration) is connected with excimer-type aggregates, which are formed by aminoxanthone molecules grouped with the help of dipole molecules of solvent or by weak hydrogen bonds between aminoxanthone molecules. The emission in band _{2 max} = 528 nm is caused by reversible changes in the 2-aminoxanthone molecule and probably is connected with an intramolecular proton transfer.     

Theoretical investigation of noncollinear phase-matched parametric four-photon amplification of ultrashort light pulses in isotropic media

The amplification of light signals (angular frequency _S in some isotropic media (D₂O, fused silica, and Schott type SF10 glasses) by noncollinear phase-matched parametric four-photon interaction ₁+_2S+₁ is studied theoretically. Computer simulations are carried out for fundamental and second-harmonic pump pulses of a mode-locked Nd: glass laser. Degenerate interaction (wavelength ₁=₂=1054nm or 527 nm) and nondegenerate interaction (₁=1054nm, ₂=527 nm are considered. Characteristic phase-matching parameters and gain parameters versus wavelength are determined. Limitations by spectral bandwidth, optical absorption, optical damage, self-phase modulation, self-focusing and stimulated Raman scattering are analysed.     

Soliton mass and surface tension in the (λ|Ø|4)2 quantum field model

The spectrum of the mass operator on the soliton sectors of the anisotropic (|ø|⁴)₂—and the (ø⁴)₂—quantum field models in the two phase region is analyzed. It is proven that, for small enough >0, the mass gapm _s() on the soliton sector is positive, andm _s()=0(^–1). This involves estimatingm _s() from below by a quantity () analogous to the surface tension in the statistical mechanics of two dimensional, classical spin systems and then estimating () by methods of Euclidean field theory. In principle, our methods apply to any two dimensional quantum field model with a spontaneously broken, internal symmetry group.A Sloan Foundation Fellow; Research supported in part by the U.S. National Science Foundation under Grant No. MPS 75-11864.Supported in part by the National Science Foundation under Grant No. PHY 76-17191     

On the existence of invariant,absolutely continuous measures

Let (, , ) be a measure space with normalized measure,f: a nonsingular transformation. We prove: there exists anf-invariant normalized measure which is absolutely continuous with respect to if and only if there exist >0, and , 0<<1, such that (E)< implies (f ^–k(E))< for allk0.     

Kac-Moody symmetry of generalized non-linear Schrödinger equations

The classical non-linear Schrödinger equation associated with a symmetric Lie algebra =km is known to possess a class of conserved quantities which from a realization of the algebrak []. The construction is now extended to provide a realization of the Kac-Moody algebrak[, ^–1] (with central extension). One can then define auxiliary quantities to obtain the full algebra [, ^–1]. This leads to the formal linearization of the system.     

Absolutely continuous invariant measures for one-parameter families of one-dimensional maps

Given a one-parameter familyf (x) of maps of the interval [0, 1], we consider the set of parameter values for whichf has an invariant measure absolutely continuous with respect to Lebesgue measure. We show that this set has positive measure, for two classes of maps: i)f (x)=f(x) where 0<4 andf(x) is a functionC ³-near the quadratic mapx(1–x), and ii)f (x)=f(x) (mod 1) wheref isC ³,f(0)=f(1)=0 andf has a unique nondegenerate critical point in [0, 1].     

Energy level crossings in quantum mechanics

From the eigenvalue equationH \ _n () =E _n ()\ _n () withH H ₀ +V one can derive an autonomous system of first order differential equations for the eigenvaluesE _n () and the matrix elementsV _mn () where is the independent variable. To solve the dynamical system we need the initial valuesE _n ( = 0) and \ _n ( = 0). Thus one finds the motion of the energy levelsE _n (). We discuss the question of energy level crossing. Furthermore we describe the connection with the stationary state perturbation theory. The dependence of the survival probability as well as some thermodynamic quantities on is derived. This means we calculate the differential equations which these quantities obey. Finally we derive the equations of motion for the extended caseH =H ₀ +V ₁ + ² V ₂ and give an application to a supersymmetric Hamiltonian.     

On the large-coupling-constant behavior of the Liapunov exponent in a binary alloy

We consider the usual one-dimensional tight-binding Anderson model with the random potential taking only two values, 0 and, with probabilityp and 1–p, 0<p<1. We show that the Liapunov exponent (E), E R. diverges as uniformly in the energyE. Using a result of Carmona, Klein, and Martinelli, this proves that for large enough, the integrated density of states is singular continuous. We also compute explicitly the exact asymptotics for a dense set of energies and we compare the results with numerical simulations.     

Analysis of the order parameter for uniform nearest particle system

The uniform nearest particle system (UNPS) is studied, which is a continuoustime Markov process with state space . The rigorous upper bound ^(mf) = ( – 1)/ for the order parameter ₂, is given by the correlation identity and the FKG inequality. Then an improvement of this bound ^(mf) is shown in a similar fashion; C( – 1)/|log( – 1) for >1. Recently, Mountford proved that the critical value _c=1. Combining his result and our improved bound implies that if the critical exponent exists, it is strictly greater than the mean-field value 1 in the weak sense.     

Theory of fluctuations in limit-cycles of second-order bifurcation type

The fluctuations in limit cycles of second-order bifurcation (transition from a stable to an unstable focus) are investigated near the bifurcation point _c, being an external control parameter. Two different methods are applied: a time- and space-dependent Fokker-Planck equation obtained from an ^1/2-expansion of the master equation ( being the volume) and a time- and space-dependent Langevin equation. Both methods give the same results. It is shown that the dependence of the radial correlation on ² = | – _c| and the time-behaviour of the phase correlation (ensemble dephasing) are determined by the dimensionality of space.     

New proofs of the existence of the Feigenbaum functions

A new proof of the existence of analytic, unimodal solutions of the Cvitanovi-Feigenbaum functional equation g(x) = –g(g(–x)),g(x) 1 - const|x|^r at 0, valid for all in (0, 1), is given, and the existence of the Eckmann-Wittwer functions [8] is recovered. The method also provides the existence of solutions for certain given values ofr, and in particular, forr=2, a proof requiring no computer.     

An Infinite Number of Effectively Infinite Clusters in Critical Percolation

An infinite number of effectively infinite clusters are predicted at the percolation threshold, if effectively infinite means that a cluster's mass increases with a positive power of the lattice size L. All these cluster masses increase as L _D with the fractal dimension D = d – /v, while the mass of the rth largest cluster for fixed L decreases as 1/r , with = D/d in d dimensions. These predictions are confirmed by computer simulations for the square lattice, where D = 91/48 and = 91/96.