Spectral properties of random Schrödinger operators with unbounded potentials

We investigate spectral properties of random Schrödinger operators H = - + _n()(1 + |n|) acting onl ²(Z ^d), where _n are independent random variables uniformly distributed on [0, 1].Research partially supported by a Sloan Doctoral Dissertation Fellowship and NSERC under grant OGP-0007901Research partially supported by NSF grant DMS-9101716     

Cosmological Dynamics of the Tachyon with an Inverse Power-Law Potential

We investigate tachyon dynamics with an inverse power-law potential V () ^–. We find global attractors of the dynamics leading to a dust behavior for > 2 and to an accellerating universe for 0 < 2. We study linear cosmological perturbations and we show that metric fluctuations are constant on large scales in both cases. In the presence of an additional perfect fluid, the tachyon with this potential behaves as dust or dark energy.     

α-Particle Loss from Cylindrical Magnetized DT Plasma

-particles from fusion reactions in a DT magnetized plasma are confined inside the plasma not only by magnetic fields but also by induced electric fields which are due to the different mobilities of -particles and electrons. Criteria are derived for the -confinement in a cylindrical plasma, inertially confined by a heavy liner. It is observed that the suprathermal population in the DT plasma will influence the overall dynamics.     

Upper bounds onT c for one-dimensional Ising systems

We present upper bounds on the critical temperature of one-dimensional Ising models with long-range,l/n interactions, where 1<2. In particular for the often studied case of =2 we have an upper bound onT _c which is less than theT _c found by a number of approximation techniques. Also for the case where is small, such as =1.1, we obtain rigorous bounds which are extremely close, within 1.0%, to those found by approximation methods.     

Solution of coupled differential equations of soliton theory

We present the complete set of solutions of the coupled differential equations of the form ()²=(), ² =(). Equations of this form appear in several physical situations.     

Defect annealing of alpha-particle irradiated n-GaAs

The annealing behaviour of irradiation induced defects in n-type GaAs irradiated at 300 K with 5.4 MeV alpha-particles from an americium-241 (Am-241) radio nuclide have been investigated. The annealing kinetics are presented for the alpha-particle induced defects E1–E5 detected in Organo-Metallic Vapor Phase Epitaxially (OMVPE) grown n-GaAs doped with silicon to 1.2×10¹⁶ cm^–3, these kinetics are compared to those obtained for similar defects (E1–E5) detected after electron irradiation. While defects P1 and P2 were detected after removal of the electron defects E4 and E5, respectively, a new defect labelled P0, located 0.152 eV below the conduction band, was introduced by annealing. The thermal behaviour and trap characteristics of these three defects (P0–P2) are presented. In an attempt to further characterise defects P0 and P1 a preiliminary study investigating the emission rate field dependence of these defects was conducted, it was observed that defect P0 exhibited a fairly strong field dependence while P1 exhibited a much weaker dependence.     

Nonlinear superposition for Liouville's equation in three spatial dimensions

We derive in 3+0 dimensions exact solutions of Liouville's equation ²=exp , by applying the Bäcklund transformation technique in conjunction with the principle of nonlinear superposition. The procedure, which is later extended to 3+1 dimensions, yield, as a byproduct, particular solutions of ² and ² =exp (² +² ).     

Perturbations of flows on Banach spaces and operator algebras

For automorphism groups of operator algebras we show how properties of the difference _t – ' _t are reflected in relations between the generators , . Indeed for a von Neumann algebraM with separable predual we show that if _t – '_t 0.28 for smallt, then = 0(+)°^-1 where is an inner automorphism ofM and is a bounded derivation ofM. If the difference _t – ' _t =O(t) ast ; 0, then = + and if _t – ' _t 0.28 for allt then =. We prove analogous results for unitary groups on a Hilbert space andC ₀,C ₀ ^* groups on a Banach space.This paper subsumes an earlier work of the same title which appeared as a report from Z.I.F. der Universität BielefeldWith partial support of the U.S. National Science Foundation     

Quantum Observables,Lie Algebra Homology and TQFT

Let us consider a Lie (super)algebra G spanned by T where T are quantum observables in BV formalism. It is proved that for every tensor c^... that determines a homology class of the Lie algebra G the expression c^...T...T is again a quantum observable. This theorem is used to construct quantum observables in the BV sigma model. We apply this construction to explain Kontsevich's results about the relation between homology of the Lie algebra of Hamiltonian vector fields and topological invariants of manifolds.     

Monotonicity and concavity in Coulomb systems

The eigenvalues ofH()=H ₀+H ^*, whereH ^* is an arbitrary Coulomb potential, decrease with increasing 0. Linear and parabolic bounds for the ground-state energy are presented. These bounds are applied to the biexciton and the exciton at a neutral donor.     

The Sine-Gordon field theory model at α2=8π, the non-superrenormalizable theory

We study the Sine-Gordon field theory at ²=8. We prove that the theory is renormalizable but not superrenormalizable and we show how the perturbative renormalization procedure works in this case where the interaction is not polynomial. To go beyond the perturbative results we investigate the -functional equation for this theory and discuss in what sense at ²=8 the theory is lacking the asymptotic freedom and how it is asymptotic free for ²<8 in a appropriate region of the coupling constants.     

Some results for the exponential interaction in two or more dimensions

We show that for the regularized exponential interaction :e : ind space-time dimensions the Schwinger functions converge to the Schwinger functions for the free field ifd>2 for all or ifd=2 for all such that ||>₀.Partially sponsored by the I.H.E.S. through the Stiftung Volkswagenwerk     

Relativistic alpha-particles from inelastic collisions of nuclei of24Mg in photoemulsion at 4.5 GeV/(c· nucleon)

The transverse momentum spectra and correlation characteristics of secondary -particles (fragments of the²⁴Mg nuclei) are studied. An excessive number of high-momentum -particles are found, which cannot be explained by the statistical model. It is shown that the azimuth correlations of -particles can be due to a large transverse momentum of the fragmenting residual magnesium nucleus.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 19–23, May, 1991.     

Massive spin-two particle in a gravitational field

The spin-two particle is described by a symmetric tensorh subject to the subsidiary conditionsh = h =0. Their covariant generalization and the wave equation have been obtained directly from the Eulerian variational equations by algebraic methods only. In addition to the tensor fieldh a symmetric third-rank tensor = as well as a vector fieldA have been added, neither of which enter in the final result. The Lagrangian function is taken as a linear sum of all combinations which can be constructed from these functions, as well as terms involving the curvature tensor and its two possible contractions. Variation with respect toh , andA independently gives the Euler equations. Combining the various trace equations and choice of arbitrary constants yields the subsidiary conditions, while the Euler equations themselves give the connection between the auxiliary functions and the tensorh as well as the generalization of the wave equationD D h + 2R h -R h -R h +g R h +Rh =m ² h Finally, variation with respect tog yields the energy-momentum tensor.     

Analytic study of power spectra of the tent maps near band-splitting transitions

Successive band-splitting transitions occur in the one-dimensional map x_i+1=g(x_i),i=0, 1, 2,... withg(x)=x, (0 x 1/2) –x +, (1/2 <x 1) as the parameter is changed from 2 to 1. The transition point fromN (=2ⁿ) bands to 2Nbands is given by=(2)^1/N (n=0, 1,2,...). The time-correlation function _i=x_ix₀/(x₀)²,x_i x_i–x_i is studied in terms of the eigenvalues and eigenfunctions of the Frobenius-Perron operator of the map. It is shown that, near the transition point=2, _i–[(10–42)/17] _i,0-[(102-8)/51]_i,1 + [(7 + 42)/17](–1)ⁱe^–yi, where2(–2) is the damping constant and vanishes at=2, representing the critical slowing-down. This critical phenomenon is in strong contrast to the topologically invariant quantities, such as the Lyapunov exponent, which do not exhibit any anomaly at=2. The asymptotic expression for _i has been obtained by deriving an analytic form of _i for a sequence of which accumulates to 2 from the above. Near the transition point=(2)^1/N, the damping constant of _i fori N is given by _N=2(^N-2)/N. Numerical calculation is also carried out for arbitrary a and is shown to be consistent with the analytic results.     

Feynman rules for probability amplitudes

Starting with very simple assumptions, Feynman rules for the quantum mechanical amplitudes and the associated probabilities are derived. These rules emerge as the only consistent rules for manipulating complex amplitudes assigned to processes. The probability of a process to which an amplitudex has been assigned is determined asp(x)=|x| , 0<2. If virtual processes are allowed,=2.     

Hamiltonian formulation of general relativity in terms of Dirac spinors

The Dirac spinors and matrices are used in combination with the Arnowitt-Deser-Misner formalism in order to obtain yet another formulation of Hamiltonian general relativity, together with a new form of the Gauss-Codazzi equations. The relation with Ashtekar's variables is analyzed; it is shown, for instance, that the matrices are equivalent to the electric field variable. The electric and magnetic decomposition of the gravitational field is also studie using Dirac matrices.     

The strength of Einstein's equations

The strength of Einstein's empty-space field equations is computed anew and shown to be equal to the amount of initial data required for a local solution of the equations. This same amount of initial data is shown to be precisely that required for a set of 16 unknown first-order differential equations containing 10 field variables and having six identities of second order. The 10 field variables must be functions of second order in the metric coefficients. The 16 field equationsC , = 0 whereC is Weyl's conformal tensor, are shown to have the same properties as those of the unknown equations, suggesting thatC = 0 is a satisfactory local first-order formulation of Einstein's second-order empty-space field equations.     

An inequality for partition functions with disturbed Hamiltonians

We consider a thermodynamic system consisting ofn independent subsystems. Each subsystem is described by a HamiltonianH=H ₀+ ⁱ H ₁,i=1, 2,...,n. We answer the question of how the entirety=( ¹, ²,..., ⁿ ) must be varied in order to change the total partition function and the total free energy of the system monotonically.