On the invariant sets of a family of quadratic maps

The Julia setB for the mappingz (z–)² is considered, where is a complex parameter. For 2 a new upper bound for the Hausdorff dimension is given, and the monic polynomials orthogonal with respect to the equilibrium measure onB are introduced. A method for calculating all of the polynomials is provided, and certain identities which obtain among coefficients of the three-term recurrence relations are given. A unifying theme is the relationship betweenB and -chains ± (± (± ...), which is explored for –1/42 and for with ||1/4, with the aid of the Böttcher equation. ThenB is shown to be a Hölder continuous curve for ||<1/4.Supported by NSF Grant MCS-8104862Supported by NSF Grant MCS-8002731     

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.     

Some tree-like Julia sets and Padé approximants

The Julia set B for T(z)=(z–)², the equilibrium electrostatic measure on B, the associated orthogonal polynomials, P _n, and the Padé approximants to the moment-generating function for are considered. When 02, the locations of the zeros and poles for the Padé approximant sequence (z), n=0, 1, 2, ..., are described precisely as vertices of trees of analytic arcs associated with B. For infinitely-many values of B is the closure of these trees. P ₂ ⁿ is shown to be a Chebychev polynomial on B for positive, attaining its maximum modulus at 2 ⁿ⁺¹ points of B if 2.Supported by NSF Grant MCS-8104862Supported by NSF Grant MCS-80002731     

A new rigorous upper bound for the inverse critical temperature of the two-dimensional Coulomb gas

In this paper we show how to improve the recent result _c 17.2 on the inverse critical temperature for the two-dimensional Coulomb gas at low density to get the following upper bound: _c 16.     

C λ-extended oscillator algebra and parasupersymmetric quantum mechanics

The C -extended oscillator algebra is generated by {1, a, a , N, T}, where T is the generator of the cyclic group C of order . It can be realized as a generalized deformed oscillator algebra (GDOA). Its unirreps can thus be easily exhibited using the representation theory of GDOAs and their carrier spaces show a Z-grading structure. Within its infinite-dimensional Fock space representation, this algebra provides a bosonization of parasupersymmetric quantum mechanics of order p = – 1.     

Perturbation about the mean field critical point

We consider two models that are small perturbations of Gaussian or mean field models: the first one is a double well /4⁴ — /2² perturbation of a massless Gaussian lattice field in the weak coupling limit (0, proportional to ). The other consists of a spin 1/2 Ising model with long-range Kac type interactions; the inverse range of the interaction, , is the small parameter. The second model is related to the first one via a sine-Gordon transformation. The lattice ^d has dimensiond3.In both cases we derive an asymptotic estimate to first order (in or ²) on the location of the critical point. Moreover, we prove bounds on the remainder of an expansion in or around the Gaussian or mean field critical points.The appendix, due to E. Speer, contains an extension of Weinberg's theorem on the divergence of Feynman graphs which is used in the proofs.Supported by NSF Grant # MCS 78-01885Supported by NSF Grant # PHY 78-15920     

Functional derivative for the distribution function of a Λ-nucleon pair in nuclear matter

An exact expression for the functional derivative of the distribution function of a -nucleon pair in nuclear matter is derived. An approximate expression is also derived by means of the Kirkwood superposition approximation. The latter expression is subsequently used to obtain the Euler equation for the correlation functionf(r1) of a -nucleon pair in nuclear matter.     

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.     

Singular measures without restrictive intervals

For the transformationT:[0,1][0,1] defined byT(x)=x(1–x) with 04, a is shown to exist for whichT has no restrictive intervals, hence is sensitive to initial conditions, but for which no finite absolutely continuous invariant measure exists forT.Supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University     

Energy relaxation in aϕ 4 with long range interactions

We investigate the influence of long range interactions on the relaxation behaviour of a lattice model with an on-site potential of ⁴-type and infinite range harmonic interactions. For finite number of particlesN, it is shown that the autocorrelation functions <E _n(t)E _n > of the fluctuations of the one-particle energiesE _n(t) decays exponentially. The corresponding relaxation time is proportional toN and is given by (T, N) =N₀(T). The temperature dependent time scale ₀ can explicitly be related to the dynamics of a one-particle correlator of the noninteracting system. The results are derived using Mori-Zwanzig projection formalism. The corresponding memory kernel is calculated within a mode coupling approximation and by a perturbative approach. Both results agree in leading order in 1/N. It is speculated that any interaction of range generates a timescale .     

The lateral photovoltaic effect in CdS-Cu2S heterojunction solar cell

The lateral photovoltaic effect has been observed in CdS-Cu₂S thin-film solar cells. The effect is more pronounced on the CdS side than on the Cu₂S side of the cells. On the CdS side, where the contacts were formed by soldering Cu wire by indium and then applying Ag paint, the photovoltage developed were found to increase as the point of illumination was moved towards the contact. The spectral response of photovoltage for coevaporated cells shows a peak at=0.5m (2.45 eV). But for topotaxial cells two peaks, one at=0.5m and the other at=0.65m (1.89eV) were observed. A band model has been proposed for the heat-treated optimized cells.     

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.     

On the thermodynamics of associative recall of structured patterns within a given context

Within a thermodynamic approach we study the associative recall of structured patterns, e.g. complex impressions composed of auditory and visual components, or words consisting of various letters. These words can be recognized either without context, i.e. letter by letter in an independent way, or they can be put into context by favouring a certain percentage of preferred letter combinations (meaningful words) in the Hamiltonian. Particular emphasis is put on the question, under which conditions the system can recognize the preferred words in an improved way without loosing the quality of retrieval of the remaining arbitrary letter combinations. For two-letter words we find a phase diagram depending on three parameters, i.e. the temperatureT, the ratio between the number of letters within the alphabet and the number of neurons, and the enhancement parameter for the preferred combinations. This phase diagram is very rich, e.g. for a given and 0<<1, below a critical temperatureT ₂() one can recognize both the preferred words and-with somewhat reduced error tolerance-also any arbitrary letter combination, whereas forT ₂()<T<T ₁() only the preferred words can be recognized. For <0 the role of preferred and non-preferred combinations is interchanged.     

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.     

On Wheeler's “rule of unanimity” in quantum cosmology

Counterexamples are given to Wheeler's rule of unanimity, which implies that every quantum solution for the problem of a closed universe within the framework of Einstein's field equations leads to a singularity. Bianchi IX universes filled with a perfect fluid with an equation of state of the type =(–1) (=const, between 1 and 2), quantized in the framework of the canonical scheme proposed by Lund and generalized by Demaret and Moncrief, are indeed shown to be nonsingular, apart from a set of measure zero of models including the closed Friedmann-Robertson-Walker models.     

Power law growth for the resistance in the Fibonacci model

Many one-dimensional quasiperiodic systems based on the Fibonacci rule, such as the tight-binding HamiltonianH(n)=(n+1)+(n–1)+v(n) (n),n,l ²(),, wherev(n)=[(n+1)]–[n],[x] denoting the integer part ofx and the golden mean , give rise to the same recursion relation for the transfer matrices. It is proved that the wave functions and the norm of transfer matrices are polynomially bounded (critical regime) if and only if the energy is in the spectrum of the Hamiltonian. This solves a conjecture of Kohmoto and Sutherland on the power-law growth of the resistance in a one-dimensional quasicrystal.     

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 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.     

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.     

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.