An extremal problem related to probability

Summary We consider a walk from a stateA ₁ to a stateA _n+1 in which the probability of remaining atA _i isp _i , and the probability of progressing fromA _i toA _i+1 is 1 –p _i . The probabilityW _nk of reachingA _n+1 fromA ₁ in exactlyn + k steps can then be expressed as a polynomial of degreen + k in then variablesp ₁,,p _n . We determine the maximum value ofW _nk and the (unique) choice (p ₁,,p _n ) for which this extremum occurs.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

The spectrum of a quasiperiodic Schrödinger operator

The spectrum (H) of the tight binding Fibonacci Hamiltonian (H _mn= _m,n+1+ _m+1,n + _m,n v(n),v(n)= ((n–1)), 1/ is the golden number) is shown to coincide with the dynamical spectrum, the set on which an infinite subsequence of traces of transfer matrices is bounded. The point spectrum is absent for any , and (H) is a Cantor set for 4. Combining this with Casdagli's earlier result, one finds that the spectrum is singular continuous for 16.On leave from the Central Research Institute for Physics, Budapest, Hungary     

Local products for relativistic quantum fields

A definition of local functions of generalized free fields is suggested. Their concrete form is determined in a fairly general setting by analysing realizations of locality. A huge class of new models of relativistic quantum fields results in this way.Presented at the International Conference Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 23–27, 1986.     

Semiclassical Topological Flat Laguerre Planes Obtained by Pasting Along A Circle

A new rather large family of locally compact 2-dimensional topological Laguerre planes is introduced here. This family consists exactly of those Laguerre planes which can be obtained by pasting together two halves of the classical real Laguerre plane along a circle suitably. Isomorphism classes and automorphism groups of these planes are determined. Together with [9] this gives a complete classification of all semicalssical topological flat Lguerre planes.     

Equilibrium polymerization in a solvent: Solution on the Bethe lattice

The lattice model for equilibrium polymerization in a solvent proposed by Wheeler and Pfeuty is solved exactly on a Bethe lattice (core of a Caylay tree) with general coordination numberq. Earlier mean-field results are reobtained in the limitq, but the phase diagrams show deviations from them for finiteq. Whenq=2, our results turn into the solution of the one-dimensional problem. Although the model is solved directly, without the use of the correspondence between the equilibrium polymerization model and the diluten0 model, we verified that the latter model may also be solved on the Bethe lattice, its solution being identical to the direct solution in all parameter space. As observed in earlier studies of the puren0 vector model, the free energy is not always convex. We obtain the region of negative susceptibility for our solution and compare this result with mean field and renormalization group (-expansion) calculations.     

Critical behavior of betaine arsenate

The critical exponents , and of undeuterated and partially deuterated ferroelectric betaine arsenate are estimated. Small but remarkable deviations from the tricritical values are observed. The obtained values are tested via scaling of the isothermals.At some distance aboveT _c a cross-over of the effective critical exponent to a value typically for short-range forces is observed. A suitably defined cross-over temperature range decreases with increasing deuteration and seems to vanish near the antiferroelectric phase boundary.     

Quantum spin chains: Simple models with complex dynamics

The present study highlights some of the complexities observed in the dynamical properties of one-dimensional quantum spin systems. Exact results for zero-temperature dynamic correlation functions are presented for two contrasting situations:

1. a system with a fully ordered ferromagnetic ground state;
2. a system at aT _c=0 critical point.

For both situations it is found that the exact results are considerably more complex than has been anticipated on the basis of approximate approaches which are considered to be appropriate and reliable for such situations. A still higher degree of complexity is expected for the dynamics of quantum spin systems which are nonintegrable. The paper concludes with some observations concerning nonintegrability effects and quantum chaos in spin systems.