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.     

On the geometric origin of the equation ϕ,11 — ϕ,22 = eϕ — e-2ϕ

It is shown that the equation ,11 — ,22 = e — e^-2 determines the intrinsic geometry of the two-dimensional affine sphere in the three-dimensional unimodular affine space like the sine-Gordon equation describes the metric on the surface of a constant negative curvature in the three-dimensional Euclidean space. The linear equations that determine the moving frame on the affine sphere are the Lax operators to the equation ,11 — ,22 = e — e^-2.     

Melnikov's criterion for nondifferentiable weak-noise potentials

The stationary probability density of Fokker-Planck models with weak noise is asymptotically of the form exp[–1 /(q)]. If is smooth, it satisfies a Hamilton-Jacobi equation at zero energy and can be interpreted as the action of an associated Hamiltonian system. Under this assumption, has the properties of a Liapounov function, and can be used, e.g., as a thermodynamic potential in nonequilibrium steady states. We consider systems having several attractors and show, by applying Melnikov's method to the associated Hamiltonian, that in general is not differentiable. A small perturbation of a model with differentiable leads to a nondifferentiable . The method is illustrated on a model used in the treatment of the unstable mode in a laser.     

Diffraction of a plane wave in air by a cylindrical shell with two local masses

The diffraction of a plane wave by a cylindrical shell with two inertial masses attached symmetrically (at = ±₁) to its surface is investigated. The frequency response characteristics of the scattered field are calculated for a shell equipped with diametrically opposite masses (₁ = /2) and driven only by symmetric (about the plane =0, =) modes of vibration. It is established that the level of the scattered field in the interval of frequencies corresponding to even symmetric mode orders increases upon transition from the model with a single local mass (at =) to the model with two masses (at =±/2) (i.e., with a split inhomogeneity). At the frequencies of odd modes the scattered field coincides with the field of the homogeneous shell. The directivity patterns of the scattered field of a shell with two masses (at =±/2) are determined.Architecture and Construction Institute, Nizhegorod. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 37, No. 3, pp. 289–299, March, 1994.     

A class of nonlinear relativistic partial differential equations in elementary particle theory

In mathematical approaches to elementary particle theory, the equation [² - ²/t²]=m² ;+g ³ has been of interest [1,2]; it describes a quartically self-coupled neutral scalar meson field. This paper applies the decomposition method [3-6] to obtain accurate non-perturbative timedevelopment of the field for this equation, or variations involving other nonlinear interactions, without the use of cutoff functions or truncations.     

L 2-exponential lower bounds to solutions of the Schrödinger equation

We study decay properties of solutions of the Schrödinger equation (–+V)=E. Typical of our results is one which shows that ifV=o(|x|^–1/2) at infinity or ifV is a homogeneousN-body potential (for example atomic or molecular), then ifE<0 and . We also construct examples to show that previous essential spectrum-dependent upper bounds can be far from optimal if is not the ground state.Research in partial fulfillment of the requirements for a Ph.D. degree at the University of VirginiaPartially supported by NSF grant MCS-81-01665Supported by Fonds zur Förderung der wissenschaftlichen Forschung in Österreich, Projekt Nr. 4240     

Schrödinger Operators with Empty Singularly Continuous Spectra

Let H be a semibounded perturbation of the Laplacian H ₀ in L ²( ^d ). For an admissible function sufficient conditions are given for the completeness of the scattering system (H), (H ₀). If is the exponential function and if e^{– H} is an integral operator we denote the kernel of the difference D = e^{– H} – e^{– H} ₀ by D (x, y), > 0. The singularly continuous spectrum of H is empty if^d dx ^d dy |D(x,y)| (1 + |y|²)< for some > 1. This result is applied to potential perturbations and to perturbations by imposing Dirichlet boundary conditions.     

Generalization of the relativistic string model in the geometrical approach

We present a model of a one-dimensional extended relativistic object, whose motion is defined by the requirement that its time track in Minkowski space is a surface of the constant mean curvature H. The world surface of the relativistic string is a particular case of such surfaces, namely, a minimal surface with H=0. By differential-geometry methods the theory of the proposed object moving in three-dimensional space-time is reduced to one nonlinear equation – = Hsh. In the theory under consideration, there naturally arises the pair of Lax's operators needed to solve this nonlinear equation by the inverse scattering method.     

Cyclic homology of differential operators,the Virasoro algebra and aq-analogue

We show how methods from cyclic homology give easily an explicit 2-cocycle on the Lie algebra of differential operators of the circle such that restricts to the cocycle defining the Virasoro algebra. The same methods yield also aq-analogue of as well as an infinite family of linearly independent cocycles arising when the complex parameterq is a root of unity. We use an algebra ofq-difference operators andq-analogues of Koszul and the Rham complexes to construct these quantum cocycles.     

The effect of substrate work function on work function reduction of Re/W alloy-coated impregnated cathodes

The work function reduction at a metal surface caused by adsorbed barium and oxygen is already known with the resulting expression for the overall work function being =_S–_R. Results from the Re/W alloy-coated cathodes exhibit that _R increases with substrate work function with a slope of 1.09 by linear regression. A qualitative explanation of the above results comes from considering the arrangement of barium and oxygen on the surfaces. A charge transfer model is proposed for two-dimensional barium and oxygen lying on a flat non-interacting substrate. Calculation shows that the dipole strength increases with the charge transfer below the intermediate states.     

Lorentz-symmetric solutions of nonlinear equations

Asymptotic properties of a real scalar self-interacting classical field depending on one variablez = t ²–x ² are studied. The field(z) approaches a minimum of the potentialU() for z + and a maximum forz– ifU((0)) is larger than two minima and smaller than two maxima ofU neighbouring to(0).Presented at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–19, 1981.     

Operator product expansions on the vacuum in conformal quantum field theory in two spacetime dimensions

Let ₁(x) and ₂(y) be two local fields in a conformal quantum field theory (CQFT) in two dimensional spacetime. It is then shown that the vector-valued distribution ₁(x)₂(y)|0 is a boundary value of a vectorvalued holomorphic function which is defined on a large conformally invariant domain. By group theoretical arguments alone it is proved that ₁(x)₂(y)|0 can be expanded into conformal partial waves. These have all the properties of a global version of Wilson's operator product expansions when applied to the vacuum state |0. Finally, the corresponding calculations are carried out more explicitly in the Thirring model. Here, a complete set of local conformally covariant fields is found, which is closed under vacuum expansion of any two it its elements (a vacuum expansion is an operator product expansion applied to the vacuum).Work supported by Deutsche Forschungsgemeinschaft     

Internal Lifschitz singularities of disordered finite-difference Schrödinger operators

The integrated density of states has C-like singularities, ln|k(E)–k(E _c )|=–|E–E _c |^–v/2 _c (E), with _c >0, a milder function at the edges of the spectral gaps which appear when the distribution function of the potentiald has a sufficiently large gap. The behaviour of _c nearE _c is determined by the local continuity properties ofd near the relevant edge: _c (E)=O(1) ifd has an atom and =O(ln|E–E _c |) if is (absolutely) continuous and power bounded.     

Strict inequalities for some critical exponents in two-dimensional percolation

For 2D percolation we slightly improve a result of Chayes and Chayes to the effect that the critical exponent for the percolation probability isstrictly less than 1. The same argument is applied to prove that ifL():={(x, y):x=r cos, y=r sin for some r0, or} and():=limpp _c [log(p–p _c )]^–1 log P_cr {itO is connected to by an occupied path inL()}, then() is strictly decreasing in on [0, 2]. Similarly, lim_n [–logn]^–1 logP _cr {itO is connected by an occupied path inL()() to the exterior of [–n, n]×[–n, n] is strictly decreasing in on [0, 2].     

Construction and Borel summability of planar 4-dimensional euclidean field theory

We use the methods of [1] to show that the planar part of the renormalized perturbation theory for ₄ ⁴ -euclidean field theory is Borel-summable on the asymptotically free side of the theory. The Borel sum can therefore be taken as a rigorous definition of theN limit of a massiveN×N matrix model with a +trg ⁴ interaction, hence with wrong sign ofg. Our construction is relevant for a solution of the ultra-violet problem for planar QCD. We also propose a program for studying the structure of the renormalons singularities within the planar world.     

Strong coupling in the quantum field theory with nonlocal nonpolynomial interaction

In the relativistic quantum field theory the representation for theS-matrix elements is obtained for any coupling constantsg in the case of a one component scalar field (x) with nonlocal nonpolynomial interaction _I ()=gU() when the causal function is bounded in the Euclidean region 0D _c (x _E ² D _c (0)< and the function |U(u)|1 for realu. It is proved that the two point Green function is bounded in the physical region of momenta variablep ².     

Square root singularity in the viscosity of neutral colloidal suspensions at large frequencies

The asymptotic frequency, , dependence of the dynamic viscosity of neutral hard-sphere colloidal suspensions is shown to be of the form ₀A()(_p)^-1/2, whereA() has been determined as a function of the volume fraction , for all concentrations in the fluid range, ₀ is the solvent viscosity, and _p is the Péclet time. For a soft potential it is shown that, to leading order in the steepness, the asymptotic behavior is the same as that for the hard-sphere potential and a condition for the crossover behavior to 1/_p, is given. Our result for the hardsphere potential generalizes a result of Cichocki and Felderhof obtained at low concentrations and agrees well with the experiments of van der Werffet al. if the usual Stokes-Einstein diffusion coefficientD ₀ in the Smoluchowski operator is consistently replaced by the short-time self-diffusion coefficientD _s() for nondilute colloidal suspensions.     

Correlation inequalities for two-component hypercubicϕ 4 models

A collection of new and already known correlation inequalities is found for a family of two-component hypercubic ⁴ models, using techniques of duplicated variables, rotated correlation inequalities, and random walk representation. Among the interesting new inequalities are: rotated very special Dunlop-Newman inequality _1,x ² ; _1,z ² + _2g ² 0, rotated Griffiths I inequality _{1,x 1,y} ; _1z ² 0, and anti-Lebowitz inequalityu ₄ ¹¹¹¹ >-0.     

Shell sources of stationary axisymmetric gravitational fields

A number of solutions for material shell sources of stationary axisymmetric gravitational fields are presented. Explicit solutions are found for shells lying on equipotential hypersurfaces (g _tt = const) and generating static monopole fields in prolate and oblate spheroidal coordinates (Zipoy-Voorhees fields). Numerical solutions are found for shells lying on hypersurfaces of constantg /g and generating Kerr- and Tomimatsu-Sato ( = 2) fields. The shells have minimum areas allowed by the energy conditions of Hawking and Ellis.     

Operators with singular continuous spectrum,VII. Examples with borderline time decay

We construct one-dimensional potentialsV(x) so that if onL ²(), thenH has purely singular spectrum; but for a dense setD, D implies that |,e ^-itH |C |t|^-1/2 ln(|t|) for t>2. This implies the spectral measures have Hausdorff dimension one and also, following an idea of Malozemov-Molchanov, provides counterexamples to the direct extension of the theorem of Simon-Spencer on one-dimensional infinity high barriers.This material is based upon work supported by the National Science Foundation under Grant No. DMS-9401491. The Government has certain rights in this material.