On the geometric structure underlying the eikonal equation

Given the eikonal equation _i=1 ³ (/x ⁱ ) ² =n ², we investigate the geometric structure that underlies the law of propagation of the wavefronts (x ¹,x ²,x ³) —ct=0. It turns out that Huygens' principle for the propagation of wavefronts is given in terms of a contact structure. Wavefronts are carried into wavefronts by contact transformations. As regards the wave-particle duality principle that arises in quantum mechanics, there is a natural geometric structure, a symplectic manifold (M ²ⁿ , ), which unifies Fermat's principle and the eikonal equation (Huygens' principle).On leave of absence from Institut für Angewandte Mathematik, Fachbereich Mathematik der Universität Mainz, Mainz, German Federal Republic.     

Percolation of level sets for two-dimensional random fields with lattice symmetry

Let (x),x², be a random field, which may be viewed as the potential of an incompressible flow for which the trajectories follow the level lines of . Percolation methods are used to analyze the sizes of the connected components of level sets {x:(x)=h} and sets {x:(x)h} in several classes of random fields with lattice symmetry. In typical cases there is a sharp transition at a critical value ofh from exponential boundedness for such components to the existence of an unbounded component. In some examples, however, there is a nondegenerate interval of values ofh where components are bounded but not exponentially so, and in other cases each level set may be a single infinite line which visits every region of the lattice.     

Entropy,the central limit theorem and the algebra of the canonical commutation relation

The central limit theorem of Cushen and Hudson is reformulated on the algebra of the CCR. Namely, for a gauge invariant state , the weighted convolutions _n of the central limit tend to the quasi-free reduction _Q of pointwise. It is proved that if the initial relative entropy S(, _Q ) is finite, then S( _n , _Q ) goes to 0 and so _n – _Q 0. No restriction on the dimension of the test function space is made.     

Path integrals on half-line

In the framework of path integrals we present a solution to the Schrödinger equation for a free particle confined to the half-linex > 0. A solution in question corresponds to the boundary condition (/x) (0,t)= (0,t) where is a real constant.     

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.     

Bilinear quantum Monte Carlo: Expectations and energy differences

We propose a bilinear sampling algorithm in the Green's function Monte Carlo for expectation values of operators that do not commute with the Hamiltonian and for differences between eigenvalues of different Hamiltonians. The integral representations of the Schrödinger equations are transformed into two equations whose solution has the form _a(x) t(x, y)_b(y), where _a and _b are the wavefunctions for the two related systems andt(x, y) is a kernel chosen to couplex andy. The Monte Carlo process, with random walkers on the enlarged configuration spacex y, solves these equations by generating densities whose asymptotic form is the above bilinear distribution. With such a distribution, exact Monte Carlo estimators can be obtained for the expectation values of quantum operators and for energy differences. We present results of these methods applied to several test problems, including a model integral equation, and the hydrogen atom.     

First experiment to test whether space-time is flat. II. Effects of orbit

The gyroscope in an orbiting satellite will be acted on by additional gravitational fields due to the rotation of the earth and due to the orbital velocity of the satellite. According to special relativistic gravitational theory, we deduce _L ^(S) —the gyroscope's precession rate due to the orbital velocity—and _S ^(S) —the gyroscope's precession rate due to the earth's rotation in the polar orbit case. The results are _L ^(S) = (2/3) _L ^(G) , _S ^(S) = (3/2) cos (1 - sin² cos²)^1/2 _S ^(G) , where and are the gyroscope's polar angles, and _L ^(G) and _S ^(G) are the geodetic and frame-dragging precession rates predicted by general relativity, respectively.     

Exponential decay in the stark effect

LetH=–+V+Fx ₁ withV(x ₁,x ) analytic in the first variable andV(x ₁+ia, x ) bounded and decreasing to zero asx for eacha . Let be an eigenvector of –+V with negative eigenvalue. Among our results we show that forF0, (,e ^–H ) decays exponentially at a rate governed by the positions of the resonances ofH. This exponential decay is in marked contrast to conventional atomic resonances for which power law decay is the rule.Research supported by NSF Grant No. MCS 78-00101.     

Collision Integrals for Attractive Potentials

Motivated by previous discussions of particle interactions under the Manev potential U(r)=–/r–/r ², we construct the collision integrals for attractive potentials U(r) satisfying the condition U(r) r ²– as r0 with 0. For =0, we obtain a Boltzmann-type integral with a collision law allowing spiral interactions and nonunique correspondence between impact parameter and scattering angle. For >0, an additional Smoluchowski-type coagulation integral arises. All these integrals are derived and possible applications are discussed.     

Power law growth for the resistance in the Fibonacci model

Many one-dimensional quasiperiodic systems based on the Fibonacci rule, such as the tight-binding HamiltonianH(n)=(n+1)+(n–1)+v(n) (n),n,l ²(),, wherev(n)=[(n+1)]–[n],[x] denoting the integer part ofx and the golden mean , give rise to the same recursion relation for the transfer matrices. It is proved that the wave functions and the norm of transfer matrices are polynomially bounded (critical regime) if and only if the energy is in the spectrum of the Hamiltonian. This solves a conjecture of Kohmoto and Sutherland on the power-law growth of the resistance in a one-dimensional quasicrystal.     

Asymptotic solutions of continuous-time random walks

The continuous-time random walk of Montroll and Weiss has a complete separation of time (how long a walker will remain at a site) and space (how far a walker will jump when it leaves a site). The time part is completely described by a pausing time distribution(t). This paper relates the asymptotic time behavior of the probability of being at sitel at timet to the asymptotic behavior of(t). Two classes of behavior are discussed in detail. The first is the familiar Gaussian diffusion packet which occurs, in general, when at least the first two moments of(t) exist; the other occurs when(t) falls off so slowly that all of its moments are infinite. Other types of possible behavior are mentioned. The relationship of this work to solutions of a generalized master equation and to transient photocurrents in certain amorphous semiconductors and organic materials is discussed.This work was partially supported by NSF Grant No. 28501.     

Generating Relations of the Hypergeometric Functions by the Lie Group-Theoretic Method

In this paper, the generating relations for a set of hypergeometric functions _,,,m (x) are obtained by using the representation of the Lie group SL(2,C) giving a suitable interpretation to the index m in order to derive the elements of Lie algebra. The principle interest in our results lies in the fact that a number of special cases would inevitably yield too many new and known results of the theory of special functions, namely the Laguerre, even and odd generalized Hermite, Meixner, Gottlieb, and Krawtchouk polynomials.     

Fuzzy quantum logic II. The logics of unsharp quantum mechanics

A survey of the main results of the Italian group about the logics of unsharp quantum mechanics is presented. In particular partial ordered structures playing with respect to effect operators (linear bounded operatorsF on a Hilbert space such that, 0¦F²) the role played by orthomodular posets with respect to orthogonal projections (corresponding to sharp effects) are analyzed. These structures are generally characterized by the splitting of standard orthocomplementation on projectors into two nonusual orthocomplementations (afuzzy-like and anintuitionistic-like) giving rise to different kinds of Brouwer-Zadeh (BZ) posets: de Morgan BZ posets, BZ^* posets, and BZ³ posets. Physically relevant generalizations of ortho-pair semantics (paraconsistent, regular paraconsistent, and minimal quantum logics) are introduced and their relevance with respect to the logic of unsharp quantum mechanics are discussed.     

The revival of Schrödinger's interpretation of quantum mechanics

A distinction is made between two wave functions(x) and(x), The former describing a continuous distribution of electronic matter for a single system, the latter describing the regularities in repeated experiments. The classical field(x) necessarily includes the self energy and accounts for all the radiative processes without the probability interpretation.     

Quantum theory of single events: Localized De Broglie wavelets,Schrödinger waves,and classical trajectories

For an arbitrary potential V with classical trajectoriesx=g(t), we construct localized oscillating three-dimensional wave lumps (x, t,g) representing a single quantum particle. The crest of the envelope of the ripple follows the classical orbitg(t), slightly modified due to the potential V, and (x, t,g) satisfies the Schrödinger equation. The field energy, momentum, and angular momentum calculated as integrals over all space are equal to the particle energy, momentum, and angular momentum. The relation to coherent states and to Schrödinger waves is also discussed.     

Generalized Bargmann potentials

Conditions are found for the coefficient functions of a linear ordinary differential equation of the kth order ^(k)+u₁^(k-2)+...+u_k-1=^k, when its solution has the following analytical dependence on the parameter: =exp(z _i=1 ⁿ (+a_i)(a_ja_j(Z)). The problem is closely related to the finding of n-soliton solutions of the simplest form for the periodic Toda system, corresponding to A _nand C _nseries.     

Inverse Problem for Harmonic Oscillator Perturbed by Potential,Characterization

Consider the perturbed harmonic oscillator Ty=-y+x²y+q(x)y in L²(), where the real potential q belongs to the Hilbert space H={q, xq L²()}. The spectrum of T is an increasing sequence of simple eigenvalues _n(q)=1+2n+_n, n 0, such that _n 0 as n. Let _n(x,q) be the corresponding eigenfunctions. Define the norming constants _n(q)=lim_xlog |_n (x,q)/_n (-x,q)|. We show that for some real Hilbert space and some subspace Furthermore, the mapping :q(q)=({_n(q)}₀, {_n(q)}₀) is a real analytic isomorphism between H and is the set of all strictly increasing sequences s={s_n}₀ such that The proof is based on nonlinear functional analysis combined with sharp asymptotics of spectral data in the high energy limit for complex potentials. We use ideas from the analysis of the inverse problem for the operator -ypy, p L²(0,1), with Dirichlet boundary conditions on the unit interval. There is no literature about the spaces We obtain their basic properties, using their representation as spaces of analytic functions in the disk.     

A new class of long-tailed pausing time densities for the CTRW

We present some asymptotic results for the family of pausing time densities having the asymptotic (t) property(t) [t ln¹⁺(t/T)]^–1. In particular, we show that for this class of pausing time densities the mean-squared displacement r ²(t) is asymptotically proportional to ln(t/T), and the asymptotic distribution of the displacement has a negative exponential form.     

Symbolic dynamics for the renormalization map of a quasiperiodic Schrödinger equation

A rigorous analysis is given of the dynamics of the renormalization map associated to a discrete Schrödinger operatorH onl ²(), defined byH(n)=(n+1)+(n–1)+Vf(n)(n), whereV is a real parameter,f is a certain discontinuous period-1 function, and is the golden mean. The renormalization map forH is a diffeomorphism,T, of ³, preserving a cubic surfaceS _V . ForV8 we prove that the non-wandering set of the restriction ofT toS _v is a hyperbolic set, on whichT is conjugate to a subshift on six symbols. It follows from results in dynamical systems theory that the optimally approximating periodic operators toH have spectra which obey a global scaling law. We also define a set which we call the pseudospectrum of the operatorH. We prove it to be a Cantor set of measure zero, and obtain bounds on its Hausdorff dimension. It is an open question whether the pseudospectrum coincides with the spectrum ofH.     

On Relativistic Collisional Invariants

Two simple proofs of the result that a relativistic summational invariant is a linear combination of the momentum four-vector p are given by assuming that is a continuous and differentiable function of class C ². The results can be extended to the case when is just assumed to be a generalized function.