Eigenfunctions of the Koch snowflake domain

We obtain non-tangential boundary estimates for the Dirichlet eigenfunctions _n and their gradients {_n } for a class of planar domains with fractal boundaries. This class includes the quasidiscs and, in particular, snowflake-type domains. When applied to the case when is the Koch snowflake domain, one of our main results states that {₁()} tends to or 0 as approaches certain types of boundary points (where ₁ > 0 denotes the ground state eigenfunction of the Dirichlet Laplacian on ). More precisely, let Ob (resp., Ac) denote the set of boundary points which are vertices of obtuse (resp., acute) angles in an inner polygonal approximation of the snowflake curve . Then given Ob (resp., Ac), we show that {₁()} (resp., 0) as tends to in within a cone based at . Moreover, we show that blowup of {₁} also occurs at all boundary points in a Cantor-type set. These results have physical relevance to the damping of waves by fractal coastlines, as pointed out by Sapovalet al. in their experiments on the Koch drum.Research partially supported by the National Science Foundation under Grant DMS-9207098.     

TypeN vacuum space-times as special functions in ℂ2

The members of one explicit class of functions in ² are identified with the geodetic shear-free null congruences in Minkowski's space-time. Members of a second explicit class are identified with the type-N vacuum space-times with twist-free rays. These two classes are special subclasses from a larger class of functions associated with the type-N space-times. This larger class is characterized in the following way: If and are holomorphic variables in ², thenu (, , ), a function holomorphic in, belongs to the class provided the function u/ u satisfies the tangential Cauchy-Riemann equation for an antiholomorphic function on the 3-surface whereu (, , ) has real values.This work was supported in part by NSF grant No. MPS74-14191-A01.     

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     

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.     

Approximate ground state in a model with spontaneous symmetry breakdown

The spontaneous symmetry breakdown is treated by means of a variational approach. Use is made of coherent states of Glauber and of pairing states of BCS-type as the translationally invariant vacuum states for the discussion of the real scalar field ⁴ withm ₀ ² 0. The first type of trial states reproduces the usual approach to spontaneous symmetry breakdown (-) in the tree approximation (which is possible only form ₀ ² <0), while the second type of trial states offers the possibility of spontaneous symmetry breakdown even form ₀ ² =0.     

Effective Potential for \mathcal{P}\mathcal{T}-Symmetric Quantum Field Theories

Recently, a class of -invariant scalar quantum field theories described by the non-Hermitian Lagrangian = () ² +g ² (i) was studied. It was found that there are two regions of . For <0 the -invariance of the Lagrangian is spontaneously broken, and as a consequence, all but the lowest-lying energy levels are complex. For 0 the -invariance of the Lagrangian is unbroken, and the entire energy spectrum is real and positive. The subtle transition at =0 is not well understood. In this paper we initiate an investigation of this transition by carrying out a detailed numerical study of the effective potential V _eff (_c) in zero-dimensional spacetime. Although this numerical work reveals some differences between the <0 and the >0 regimes, we cannot yet see convincing evidence of the transition at =0 in the structure of the effective potential for -symmetric quantum field theories.     

Correlation function formulas for some infrared asymptotic-free scalar field lattice models via the block renormalization group

We consider a class of scalar field lattice models with action 1/2()+V(), V small. After n block renormalization group transformations, new formulas are obtained for the finite lattice generating and correlation functions. For some infrared asymptotic-free models in the thermodynamic and n limits, the formulas for correlation functions are especially simple, isolate the correct dominant long-distance behavior, and can be used to control the subdominant contributions.     

Systematic study of muon transfer to sulphur dioxide

In a systematic study of the transfer process to sulphur dioxide, in seven different H₂ + SO₂ gas mixtures, the time spectra of the muonic sulphur X-rays yield muon transfer rates to the SO₂ molecule, deduced from the lifetimes of the p atoms, which agree all well with each other. The muonic oxygen time spectra show an additional structure as if p atoms of another kind were present. Reduced transfer rates_O are reproducible if one uses the model of ephemeral p atoms. The intensity ratios between the different kinds of p atoms are also discussed in the framework of this model and the one of black and white p atoms.     

Existence of three phases for aP(ϕ)2 model of quantum field

In the two-dimensional model of the quantum field theory with lagrangean density :()²–(–)^{21/2 4}–⁶: there exist (at least) three different phases for small and some ().     

On the Nature of the “Low-Temperature Anomaly” in Metal–Semiconductor Schottky-Barrier Contacts

It is shown that the low-temperature anomaly of the I-V characteristics of homogeneous metal–semiconductor Schottky-barrier contacts (an increase in the ideality factor n and a decrease in the barrier height _bm, measured using the saturation current, with decreasing temperature, as well as the fact that their product _bn n_bm coincides very closely with the actual barrier height _b(V)) is a consequence of two factors: (1) the nonlinear dependence of the actual barrier height on the bias, resulting in an increase in the ideality factor with increasing bias voltage (current) and (2) measurement of the parameters n and _bm for the same current for all temperatures. A new expression for the flat-band barrier height _bf is derived.     

Matrix field theory for disordered local pairing superconductors with two glass order parameters and spin-flip pairbreaking processes

We derive the exact matrix field theory for a replicated grassmannian representation of a local pairing superconducting disorder ensemble including three superconducting order parameters and the spin-flip pairbreaking mechanism. Disorder is assumed to be gaussian distributed. We find by exactly solving the saddle-point equation the criterion for a vanishing gap ^–1 + ^–1 , where denotes the averaged superconducting order parameter, ^–1 the spin-flip scattering rate, and ^–1 the scattering rate corresponding to correlations of Re(–). Taken at =0, our field theory, which is exact in all orders of ^–1 , contains new terms in addition to those of theO( ^–1 ) model derived by Efetov et al. Our formulation transfers correctly to all orders the invariances of the action into symmetries of the matrix field theory. The saddle point approximation is outlined and it is shown how singular corrections to the saddle point density of states arise atE _F in a gapless superconductor. Finally singular corrections in the two particle propagator, the density correlation function and the conductivity are calculated for =0 in one loop order. It turns out that these corrections can be entirely expressed by those of the single particle density of states.     

ПРИГОТОВЛЕНИЕ И ХАРА КТЕР ПОВЕРХНОСТИ ПЛАСТИНОК AgBr

AgBr . , .     

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.     

Standpoint cosmology

An unorthodox cosmology is based on a notion of standpoint, distinguishing past from future, realized through Hilbert-space representation of the complex conformai group for 3+1spacetime and associated coherent states. Physical (approximate) symmetry attaches to eight-parameter complex Poincaré displacements, interpretable as growth of standpoint age (one parameter), boost of matter energy-momentum in standpoint rest frame (three parameters) and displacement of matter location in a compact U(1)O(4)/O(3) spacetime attached to standpoint (four parameters). An initial condition (at big bang) is characterized by a huge dimensionless parameter that breaks dilation invariance. Four major length scales are recognized, called Planck, particle, lab, and Hubble, with separations controlled by ; all physical concepts, including spacetime, depend on wideness of scale separation.     

Determination of random variations in ionosphere height from phase fluctuations of VLF signals

Conclusions A feature of our problem of restoration of fluctuations in ionosphere height is that the experimental data(t, _k) obtained at a fixed reception point are functions of time, whereas the function ₁(u, y₂) to be restored is a function of the coordinates. If we use the assumption that the irregularities migrate transversely, coordinate y₂ can be exchanged for time t. Restoration with respect to the second coordinate u=x_i–x₀/2 is in effect obtained by using data(t, _k) on some set of carrier frequencies.The resultant solution of the restoration problem is in the form of an expansion of ₁(u, y₂) in known functions determined from the observed data(t, _k). We have evaluated the solution accuracy, which depends on the overall power signal-to-noise ratio at all frequencies used. We have demonstrated that the restoration algorithm contains an optimum number (with respect to accuracy) of coefficients to be evaluated.Khar'kov Aviation Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 20, No. 8, pp. 1138–1145, August, 1977.     

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.     

The infinite coupling limit and the asymptotics of coefficients of the perturbation expansion

For some functions in quantum field theory and quantum mechanics, presented by the asymptotic expansion in coupling constant g, the method for finding the approximate asymptotics when g is proposed.Such an estimation requires the first exact terms of the perturbation theory, their high-order asymptotics and supposition on the Borel summability. This method for the Gell-Mann-Low function (g) in the (4²/3)g ₍₄₎ ⁴ model gives approximately (g) 1.8 g ².     

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.     

Zeta functions and transfer operators for piecewise monotone transformations

Given a piecewise monotone transformationT of the interval and a piecewise continuous complex weight functiong of bounded variation, we prove that the Ruelle zeta function (z) of (T, g) extends meromorphically to {z<^-1} (where =lim g^°T^n-1...g^°Tg ^1/n ) and thatz is a pole of if and only ifz ^–1 is an eigenvalue of the corresponding transfer operator L. We do not assume that L leaves a reference measure invariant.Research partially supported by the Fonds National Suisse     

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.