Effective stimulated emission and excited-state absorption cross-section spectra of para-phenylene-ethynylene polymers

The effective stimulated emission cross-section spectra, em,eff(), the stimulated emission cross-section spectra, em(), and the excited-state absorption cross-section spectra, ex()=em()–em,eff(), of the conjugated polymers poly(2,5-dioctadecyloxy-paraphenylene-ethynylene-co-2,5-thienyl) (OPT) and poly(2,5-dioctadecyloxy-paraphenylene-ethynylene-co-2,5-pyridinyl) (OPP) in tetrahydrofuran are determined in the fluorescence region. em,eff() determined by a fluorescence amplification technique is found to be small compared to em() indicating that ex() is nearly as large as em(). The laser excitation to the wide inhomogeneously broadened S1-absorption band leads to the formation of spectrally structured inhomogeneously broadened lattice relaxed exciton conformations extending on average over two monomer units which are thought to be responsible for upper-to-lower-state stimulated emission and lower-to-upper-state (excited-state) absorption.     

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.     

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.     

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.     

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     

Painlevé test and energy level motion

From the eigenvalue H|_n()=E_n() |_n(), where HH₀+V, one can derive an autonomous system of first-order differential equations for the eigenvaluesE _n() and the matrix elements V_mn(), where is the independent variable. We perform a Painlevé test for this system and discuss the connection with integrability. It turns out that the equations of motion do not pass the Painlevé test, but a weaker form. The first integrals are polynomials and can be related to the Kowalewski exponents.     

On the spectrum of the linear transport operator with generalized boundary conditions

The spectrum of the linear transport operator under generalized boundary conditions proposed by Belleni-Morante (1970 a) contains in the right half-plane (|Re>–1) only a finite number of real eigenvalues. Its essential spectrum consists of a fan of continuous spectrum lines with apex at =–1.     

Extremal invariant states with a spectrum condition and Borchers' theorem

If the energy spectrum of an extremal invariant state is not the whole real line, it is shown that is either pure or uniquely decomposed into mutually disjoint pure states in the way that =^-1 F _{0 t} dt where is a pure state satisfying = with >0. Next we give a slightly generalized version of Borchers' theorem [1] on the innerness of some automorphism group of a von Neumann algebra with a spectrum condition.     

Rigorous Spectral Analysis of the Metal–Insulator Transition in a Limit-Periodic Potential

We consider the limit-periodic Jacobi matrices associated with the real Julia sets of f (z)=z ²– for which [2, ) can be seen as the strength of the limit-periodic coefficients. The typical local spectral exponent of their spectral measures is shown to be a harmonic function in decreasing logarithmically from 1 to 0.     

Stability domain of coherent laser waves

The constant-amplitude solutions with wavelength of a semiclassical laser model exhibit three different instabilities in the (, )-plane for pumpparameters above threshold.     

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.     

Fractal drums and then-dimensional modified Weyl-Berry conjecture

In this paper, we study the spectrum of the Dirichlet Laplacian in a bounded (or, more generally, of finite volume) open set R ⁿ (n1) with fractal boundary of interior Minkowski dimension (n–1,n]. By means of the technique of tessellation of domains, we give the exact second term of the asymptotic expansion of the counting functionN() (i.e. the number of positive eigenvalues less than ) as +, which is of the form ^/2 times a negative, bounded and left-continuous function of . This explains the reason why the modified Weyl-Berry conjecture does not hold generally forn2. In addition, we also obtain explicit upper and lower bounds on the second term ofN().     

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.     

Purely absolutely continuous spectrum for almost Mathieu operators

Using a recent result of Sinai, we prove that the almost Mathieu operators acting onl ²(), (l _Y, )(n) = (l+1)+(l–)+ cos(n+) (n) have a purely absolutely continuous spectrum for almost all a provided that is a good irrational and is sufficiently small. Furthermore, the generalized eigen-functions are quasiperiodic.     

Positive resonances and the limiting amplitude principle

We define the positive resonance points of self-adjoint operators without using the analytical continuation of corresponding resolvents and show that the limiting amplitude principle for the abstract wave equation does not take place in general, if ² = , where is the disturbing frequency and is the resonance point. The asymptotics of corresponding solutions as t are obtained, which imply the growth of the oscillation amplitude as t , 0<<1, or as ln t, t .     

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.     

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.     

The inelastic reactions πp→λ(⌆) K with allowance for the contribution of branch cuts

The method of complex angular moments is used to analyze the experimental data on the inelastic reactions p ()K, taking into account branch cuts in the j-plane in the eikonal approximation. An optical model for the backward scattering is considered. Agreement with experiment is obtained in the region of small angles for the reactions (⁰)K⁰. In the case of ^–P K⁰(^–K⁺) processes, agreement is obtained with the experimental results for large (180) scattering angles.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 26–30, August, 1972.     

En">Characters and Composition Factor Multiplicities for the Lie Superalgebra $${\mathfrak{g}}{\mathfrak{l}}$$ ( m / n )

The multiplicities a of simple modules L in the composition series of Kac modules V lambda for the Lie superalgebra (m/n ) were described by Serganova, leading to her solution of the character problem for (m/n ). In Serganova's algorithm all with nonzero a are determined for a given this algorithm, turns out to be rather complicated. In this Letter, a simple rule is conjectured to find all nonzero a for any given weight . In particular, we claim that for an r-fold atypical weight there are 2r distinct weights such that a = 1, and a = 0 for all other weights . Some related properties on the multiplicities a are proved, and arguments in favour of our main conjecture are given. Finally, an extension of the conjecture describing the inverse of the matrix of Kazhdan–Lusztig polynomials is discussed.     

One-dimensional random walk with self-interaction

A standard random walk on a one-dimensional integer lattice is considered where the probability ofk self-intersections of a path =(0, (1),..., (n) is proportional toe ^–k . It is proven that for <0,n ^–1/3(n) converges to a certain continuous random variable. For >0 the formulas are given for the asymptotic Westerwater velocity of a generic path and for the variance of the fluctuations about the asymptotic motion.