Stochastic Schrödinger operators and Jacobi matrices on the strip

We discuss stochastic Schrödinger operators and Jacobi matrices with wave functions, taking values in ^l so there are 2l Lyaponov exponents ₁..._l0 _l+1..._2l =–₁. Our results include the fact that if ₁=0 on a set positive measure, thenV is deterministic and one that says that {E|exactly 2j 's are zero} is the essential support of the a.c. spectrum of multiplicity 2j.Research partially supported by USNSF under grant DMS-8416049     

Inequalities for the Schattenp-norm. III

We present some inequalities for the Schattenp-norm of operators on a Hilbert space. It is shown, among other things, that ifA is an operator such that ReAa0, then for any operatorX, AX+XA* _p 2aX _p . Also, for any two operatorsA andB, A–B ₂ ² +A*B* ₂ ² 2A–B ₂ ² .     

Continuity and relative hamiltonians

Let ( _n ) _n 1 be a norm convergent sequence of normal states on a von Neumann algebraA with _n . Let (k _n ) _n1 be a strongly convergent sequence of self-adjoint elements ofA withk _n k. It is shown that the sequence of perturbed states converges in norm to . A related result holds forC ^*-algebras. A counter-example is provided to show that it is not sufficient to assume weak convergence of ( _n ) _n 1 even whenk _n=k for alln. However, conditions are given which, together with weak convergence, are sufficient. Relative entropy methods are used, and a relative entropy inequality is proved.     

Convergence of Nelson fiffusions

Let _t, _t ⁿ ,n1, be solutions of Schrödinger equations with potentials form-bounded by –1/2 and initial data inH ¹( ^d ). LetP, P ⁿ ,n1, be the probability measures on the path space =C(₊, ^d ) given by the corresponding Nelson diffusions. We show that if { _t ⁿ } _n1 converges to _t inH ¹( ^d ), uniformly int over compact intervals, then converges to in total variation t0. Moreover, if the potentials are in the Kato classK _d , we show that the above result follows fromH ¹-convergence of initial data, andK _d -convergence of potentials.     

A study of the ϱ0 meson polarization in π− p reaction at 5 GeV/c

Experimental data on ⁰ meson polarization in ^– p reaction at 5 GeV/c are presented. Change in ⁰ meson polarization for mesons produced at smallP ² and atP ² 0·3 or produced in backward direction in CMS is demonstrated. Natural explanation of these phenomena is one pion exchange mechanism and mechanism of quark-antiquark annihilation as observed for ⁰ meson production in¯pp reactions.The authors are indebted to Ján Piút, Richard Lednický, V. S. Rumiancev and N. K. Koutsidi for discussions and critical comments.     

Duality and absence of locally generated superselection sectors for CCR-type algebras

We isolate an abstract algebraic property which implies duality in all locally normal, irreducible representations of a quasilocalC*-algebra if it holds together with two more specific conditions. All these conditions holding for the CCR-algebra ind2 space time dimensions duality follows for representations of the two-dimensional CCR-algebra generated by pure Wightman states ofP()₂-theories. We then show that algebras of this kind have no nontrivial locally generated superselection sectors which ford3 yields a first approximation to a quantum analogue of Derrick's theorem.Supported by Science Research Council     

On the Haar measure of the quantumSU(N) group

We prove that the Haar state associated to the compact matrix quantum groupSU (N) is faithful for ]–1,1[,0, and anyN2.     

On the relative entropy

ForA any subset of () (the bounded operators on a Hilbert space) containing the unit, and and restrictions of states on () toA, ent _A (|)—the entropy of relative to given the information inA—is defined and given an axiomatic characterisation. It is compared with ent _A ^A (|)—the relative entropy introduced by Umegaki and generalised by various authors—which is defined only forA an algebra. It is proved that ent and ent ^S agree on pairs of normal states on an injective von Neumann algebra. It is also proved that ent always has all the most important properties known for ent ^S : monotonicity, concavity,w* upper semicontinuity, etc.     

The proper formula for relative entropy and its asymptotics in quantum probability

Umegaki's relative entropyS(,)=TrD (logD –logD ) (of states and with density operatorsD andD , respectively) is shown to be an asymptotic exponent considered from the quantum hypothesis testing viewpoint. It is also proved that some other versions of the relative entropy give rise to the same asymptotics as Umegaki's one. As a byproduct, the inequality TrA logAB TrA(logA+logB) is obtained for positive definite matricesA andB.     

Positron annihilation anomaly in the high-temperature superconductor YBa2Cu3O7−x

We have measured the ac susceptibility of a wire with a Nb core (1.27 mm diam.) and a Cu cladding (0.37 mm thickness) atT50 K andB0.1 mG. Due to its proximity to Nb, the Cu becomes fully superconducting. From the data we find a breakdown fieldH _b =1.2 (mG) and a coherence length =2.2T ^–1/2 (m) for the Cu, as well as a field penetration depth -34T ^1/2 (m) at the Cu/Nb interface.     

Cyclotron instabilities of a two-component electron plasma

A study is made of instability at frequencies close to the electron cyclotron frequency _B and its multiples, subject to the presence of two different groups of electrons. It is shown that a mixture of hot and cold electrons ( _ph ² _pc ² ) in the region of frequencies s _B, s2 can be unstable with respect to waves of the flute type (k _z=0) with maximum increment _max ( _ph/_pc). _B, if there exists an interval of transversal velocities in whichF/ >0. When the curvature of the magnetic field is taken into account, even waves with _B can be unstable in such a plasma. The effect of spatial inhomogeneity of the hot component on flute-type instability and on two-beam cyclotron instability is also examined.The author extends his thanks to A. B. Mikhajlovskij for his valuable comments and discussions.     

Passive states and KMS states for general quantum systems

We characterize equilibrium states of quantum systems by a condition of passivity suggested by the second principle of thermodynamics. Ground states and -KMS states for all inverse temperatures 0 are completely passive. We prove that these states are the only completely passive ones. For the special case of states describing pure phases, assuming the passivity we reproduce the results of Haag et al.     

The ground state energy of a Bose gas with Coulomb interaction II

LetH _N be the quantum mechanical Hamiltonian for a neutral system of 2_N charged particles, each of unit charge. The HamiltonianH _N is assumed to act on wave functions inL ²(^6N ) which satisfy Bose statistics. It is shown that if the kinetic energy of is sufficiently small, then |H _N |–CN ^7/5 for some universal constantC.Research supported by U.S. National Science Foundation Grant DMS 8600748     

Instantons in aU(1) lattice gauge theory: A Coulomb dipole gas

We study the decompositionA=A _I +A _SW of aU(1) lattice gauge field into instanton and spin wave parts. The action also decomposes,A=A _I +A _SW +R. HereA _I is a Coulomb dipole gas,A _SW is a zero mass free field, andR is a higher order remainder. We studyA _I in detail, ford4, in the dilute gas case (which corresponds to the low temperature limit of the gauge field theory). We establish the leading behavior of the free energy:f ^–d a. Here is the lattice spacing,a is a geometrical constant and is an activity defined in terms of a small number of instanton configurations. Our methods suggest the absence of screening in the dilute dipole gas,d4, in contrast to Debye screening for the dilute monopole gas.Supported in part by the National Science Foundation under Grant PHY 76-17191Supported in part by the National Science Foundation under Grant PHY 75-21212     

Entropy andp-particle observables. I. The general program

An information-theoretic notion of entropy is proposed for a system ofN interacting particles which assesses an observer's limited knowledge of the state of the system, assuming that he or she can measure with arbitrary precision all one-particle observables and correlations involving some numberp of the particles but is completely ignorant of the form of any higher-order correlations involving more thanp particles. The idea is to define a generic measure of entropyS[ ] = –Tr log for an arbitrary density matrix or distribution function , and then, given the trueN-particle, to define a reduced _R ^P which reflects the observer's partial knowledge. The result, at any timet, is a chain of inequalitiesS[ _R ¹ ]S[ _R ² ]...S[ _R ^N ]S[], with true equalityS[ _R ^p ]=S[ _R ^p+1 ] if and only if the true factorizes exactly into a product of contributions involving all possiblep-particle groupings. It follows further than (1) if, at some initial timet ₀, the true factorizes in this way, thenS[ _R ^p (]S[ _R ^p (t ₀)] for all finite timest>t ₀, with equality if and only if the factorization is restored, and (2) the initial response of the system must be to increase itsp-particle entropy.     

Minimally classifying relative equilibria

Estimates on a minimal classification of relative equilibria in the planarn-body problem of celestial mechanics have been announced in [1], [2]. Our main theorem asserts that these estimates are actually met for anyn3 on an open set in IR _inf+ ^supn . For anyn4, this open set is proper.Research supported in part by NSF Grant MCS 13-08818 A03.     

Learning withQ-state clock neurons

We study the optimal learning capacity for neural networks withQ-state clock neurons, i.e. the states arecomplex numbers with magnitude 1 and azimuthal anglesn·2/Q, withn=0, 1, ...,Q–1. Performing a phase space analysis, the learning capacity _c for given stability can be expressed by means of a double-integral with a simple geometrical interpretation, which for vanishing reduces to _c (Q) = 4Q/(3Q–4), forQ3. Then we define a training algorithm, which generalizes the well-known AdaTron algorithm fromQ=2 toQ3 and converges very fast to the network with optimal stability, if the numberp of random patterns to be learned is smaller than _c (Q). Finally, in the conclusions, we also give hints on applications for image recognition and in a note added in proof we generalize some results to Potts model networks.     

Phase diagrams of the semi-infiniteZ(q) models by real space renormalization-group method

We investigate the three-dimensional semi-infiniteZ(q) models using an infinitesimal Migdal-Kadanoff method. A rich variety of phase diagrams is obtained. The massless spin-wave phase which appears in the infinite two-dimensionalZ(q) models on the Clock line between the disordered and ferromagnetic phase forqq _c is also present in the semi-infinite system on the surface when the bulk is disordered. We also observe that if the bulk is in the phase to which the symmetry is engendered by a subgroupZ(p), such thatp, the surface of the system is in the same phase or in a less symmetrical phase to which the symmetry is engendered by a subgroupZ(m) ofZ(p) such thatp=m (mp) with an integer number satisfying 1p. The case =p corresponds to the least symmetrical phase. Since the infiniteZ(q) models exhibit a rich variety of phase transitions and multicritical points, the semi-infinite models present new ordinary, extraordinary, surface and special phase transitions which do not occur in the semi-infinite Ising-like systems. As theZ(q) model transforms into theX–Y model whenq, we have deduced the phase diagram of the semi-infiniteX–Y model. It is qualitatively similar to the phase diagram of the semi-infinite Ising model.     

A rigorous block spin approach to massless lattice theories

The renormalization group technique is used to study rigorously the ()⁴ perturbation of the massless lattice field in dimensionsd2. Asymptoticity of the perturbation expansion in powers of is established for the free energy density. This is achieved by using Kadanoff's block spin transformation successively to integrate out high momentum degrees of freedom and by applying ideas previously used by Gallavotti and Balaban in the context of the ultraviolet problems. The method works for arbitrary semibounded polynomials in and .Supported in part by the National Science Foundation under Grant No. PHY 79-16812On leave from Department of Mathematical Methods of Physics, University, PL-00-682 Warsaw, Poland     

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