Index of content:
Volume 27, Issue 7, July 1986

Separation of variables on n‐dimensional Riemannian manifolds. I. The n‐sphere S _{ n } and Euclidean n‐space R _{ n }
View Description Hide DescriptionThe following problem is solved: What are all the ‘‘different’’ separable coordinate systems for the Laplace–Beltrami eigenvalueequation on the n‐sphere S _{ n } and Euclidean n‐space R _{ n } and how are they constructed? This is achieved through a combination of differential geometric and group theoretic methods. A graphical procedure for construction of these systems is developed that generalizes Vilenkin’s construction of polyspherical coordinates. The significance of these results for exactly soluble dynamical systems on these manifolds is pointed out. The results are also of importance for the analysis of the special functions appearing in the separable solutions of the Laplace–Beltrami eigenvalueequation on these manifolds.

An inverse problem for multidimensional first‐order systems
View Description Hide DescriptionAn inverse problem associated with N first‐order equations in n+1 dimensions, n>1, is considered: Given appropriate inverse data T reconstruct the potential q(x _{0},x), where q is an N×N off‐diagonal matrix. Although q depends on n+1 variables, it turns out that T depends on 3n−1 variables. This necessitates imposing certain constraints on T, i.e., T must be suitably characterized. The characterization problem for T is solved explicitly. Furthermore, the problem of reconstructing q is reduced to one for reconstructing a 2×2 matrix potential in two dimensions. The inverse data needed for the reduced problem are obtained in closed form from T. A method for solving two‐dimensional inverse problems has recently appeared in the literature.

A class of solvable second‐order ordinary differential equations with variable coefficients
View Description Hide DescriptionThis paper treats the problem of solving the second‐order ordinary differential equations with variable coefficients of the form d ^{2} x/d t ^{2}+( q _{1}(t)+λq _{2}(t)) x=0. It is shown that if the initial equationd ^{2} x/d t ^{2}+q _{1}(t)x=0 is in analytically solvable form and q _{2}(t) is the inverse square function of a solution for the nonlinear auxiliary equation 1/2 x d ^{2} x/d t ^{2}− 1/4 (d x/d t)^{2}+q _{1}(t)x ^{2} =1, there are exact solutions. Using an inner relationship between solutions for the initial equation and the auxiliary equation, an infinite sequence of analytically solvable differential equations is constructed step by step. Typical examples of such a sequence are shown.

Wave equations with the characteristic propagation property
View Description Hide DescriptionWithin the class of second‐order linear self‐adjoint wave equations in 1+1 dimensions, an explicit construction is given of probably all those with the characteristic propagation property, that is, those whose solutions are without tails.

Linearization of the Hamilton–Jacobi equation
View Description Hide DescriptionThrough a canonoid transformation the integration for the Hamilton–Jacobi equations is transformed into a two step procedure: the first being a linear problem and the second a quasilinear one. Examples are given.

Characteristic functional structure of infinitesimal symmetry mappings of classical dynamical systems. III. Systems with cyclic variables
View Description Hide DescriptionThis paper is a continuation of previous papers I and II with similar titles [J. Math. Phys. 2 6, 3080, 3100 (1985)]. In those papers a theory was developed that described the characteristic functional structures of infinitesimal symmetry mappings of systems of first‐ or second‐order dynamical equations. Now an investigation is made of how cyclic variables of the dynamical equations affect the symmetry equations and thereby propagate through the theory to influence the form of the characteristic functional structure of the symmetries. These special symmetries, which have a particularly simple form, are characterized by infinitesimal point mappings in which only cyclic coordinates are varied, with the variation essentially determined by c o n s t a n t s o f m o t i o n of the dynamical system. For Lagrangian systems with cyclic coordinates these special symmetry mappings include the well‐known Noether symmetries characterized by c o n s t a n t variation of the cyclic coordinates.

Mean power reflection from a one‐dimensional nonlinear random medium
View Description Hide DescriptionA model of wave propagation in a slab 0≤x≤L of a n o n l i n e a r r a n d o m medium is considered. The index of refraction is k[1+ε̃(x,w‖u ^{ε}(x,L,w)‖^{2})]^{1} ^{/} ^{2}, where ε̃(x,α)=εm(x)+ε^{2}[( n(x) +iδ(x))α+θ(x)+iγ(x)], with ε a small parameter, m,n,δ,θ,γ suitable stochastic processes,u ^{ε}(x,L,w) the wave field, and w≥0 the intensity of nonlinearity. The m e a n r e f l e c t e d p o w e r is evaluated from a certain nonlinear partial differential equation satisfied by the reflection coefficientR ^{ε}(L,w). An infinite system of ordinary differential equations for the coefficients R ^{ε} _{ n }(L), n=0,1,2,..., in the expansion of R ^{ε}(L,w) in powers of w, is then derived, and the infinitesimal generator for the process [R _{0} R ^{*} _{0} R _{1} R ^{*} _{1}], R _{ n }≡R ^{0} _{ n }(ε^{2} L), is obtained, in the d i f f u s i o n l i m i t ε→0, L→∞, ε^{2} L=const. This allows us to compute 〈‖R‖^{2}〉≂〈‖R _{0}‖^{2}〉 +2w Re〈R _{0} R ^{*} _{1}〉 as a function of ε^{2} L. In the lossless case, δ=γ=0, there is n o correction due to the nonlinearity, in such a limit, and this remains true at least up to the order O(w ^{2}). Some effects can be observed when dissipation (δ>0, γ>0) is taken into account. Numerical results are obtained and plots are given.

Formulation of the inverse problem for the reduced wave equation in momentum space
View Description Hide DescriptionThe direct and inverse problem associated with the reduced wave equation expressed in momentum space (Fourier transform space of the spatial variable) is considered. It is shown that the right inverse of the scattering operator associated with complex scattered field amplitude T(k’,k) exists for the case where the index of refractionn(x) is real and satisfies certain smoothness conditions. A quadratic integral equation involving only T(k’,k) is obtained that represents a necessary and sufficient condition for T(k’,k) to be a complex scattered field amplitude associated with a real index of refraction. For the actual physical inverse problem where only on‐shell (‖k’‖=‖k‖) values of T(k’,k) are known, the inverse problem involves solving this nonlinear system for off‐shell data from on‐shell data. Several other nonlinear systems that can be used are derived. Once T(k’,k) is known for k’,k, n(x) is readily obtained.

Discrete quantum mechanics
View Description Hide DescriptionA discrete model for quantum mechanics is presented. First a discrete phase space S is formed by coupling vertices and edges of a graph. The dynamics is developed by introducing paths or discrete trajectories in S. An amplitude function is used to compute probabilities of quantum events and a discrete Feynman path integral is presented. Many of the results can be formulated in terms of transition probabilities and unitary operators on a Hilbert spacel ^{2}(S).

Transition probability spaces
View Description Hide DescriptionHilbert‐space representations of transition probability spaces are studied. The notions of superposition and the superposition principle are introduced. It is shown that, provided the superposition principle and the postulate of minimal superposition are satisfied, transition probability space can be represented by a generalized Hilbert space.

Mobility and measurements in nonlinear wave mechanics
View Description Hide DescriptionThe theory based on the nonlinear Schrödinger equation with an additional term λ(ψ̄ψ)^{α}ψ is investigated. The standard quantum mechanical interpretation of ψ is assumed at the beginning of the considerations. It turns out that every finite set of pure states can be transformed with the aid of an adequate time sequence of external potentials into a set of pairwise nearly orthogonal states. As a consequence, there exist measurements more selective than quantum ones. In particular, it is possible to discriminate between various mixtures of states that are equivalent in quantum mechanics. The possibility of existence of deterministic measurements is also discussed.

Elementary properties of a new kind of path integral
View Description Hide DescriptionA new kind of path integral is introduced. In ordinary quantum mechanics, it gives the projectors on the eigenspaces of the Hamiltonian. For parametrized systems, it represents a direct path integral version of the Diraccanonical quantization method by giving the projector on the physical space. Its properties on the most simple examples are studied. Applying it to quantum cosmology, the Hartle–Hawking wave function of the universe is recalculated.

On the summation of the Birkhoff–Gustavson normal form of an anharmonic oscillator
View Description Hide DescriptionThe classical Birkhoff–Gustavson normal form (BGNF) has played an important role in finding approximate constants of motion, and semiclassical energies. In this paper, this role is examined in detail for the well‐known anharmonic oscillatorH=1/2(p ^{2}+x ^{2}+g x ^{4}). It is shown that, with appropriate restrictions, this is the only perturbation series that preserves the period of this system. This series has a nonzero radius of convergence in contrast to the zero radius of convergence of its quantum analog, the Rayleigh–Schrödinger perturbation series. In addition, the BGNF is generated to high order, and a technique is given based on Padé approximants for summing this series. The summation of this series makes possible an accurate comparison of torus quantization energies with the known quantum energies over the entire range of quantum numbers. This example also demonstrates that divergence of the BGNF series of a Hamiltonian is not sufficient to refute its global integrability.

Joint linearization instabilities in general relativity
View Description Hide DescriptionWhen Einstein’s equations are supplemented by symmetry conditions, linearization instabilities can occur that are not present in either of the two sets of equations. The general conditions for this joint instability are investigated. This is illustrated with an example where both the Einstein equations and the flatness condition have more linearized solutions than exact solutions. In a minisuperspace model the geometrical reason for these instabilities is shown.

The static, cylindrically symmetric strings in general relativity with cosmological constant
View Description Hide DescriptionThe static, cylindrically symmetric solutions to Einstein’s equations with a cosmological term describing cosmic strings are determined. The discussion depends on the sign of the cosmological constant.

Cartan ideal, prolongation, and Bäcklund transformations for Einstein’s equations
View Description Hide DescriptionEinstein’s equations in the Newman–Penrose formalism for vacuum, vacuum with cosmological constant, and electrovacuum fields are expressed as Cartan ideals. Two different prolongations of these ideals are obtained. These two types of prolonged ideals generalize previous prolongations for vacuum fields to vacuum with cosmological constant and electrovacuum fields. Some Bäcklund transformations are obtained for vacuum, vacuum with cosmological constant, and electrovacuum fields. These Bäcklund transformations include the generalized Kerr–Schild (GKS) transformation, and a two‐parameter generalization of the GKS transformation. GKS transformations are studied in detail. Expressions for the transformation of Newman–Penrose quantities are given and algebraic properties are discussed. It is shown that the GKS transformation cannot give algebraically general and asymptotically flat vacuum and electrovacuum space‐time metrics.

Statistical mechanics approach in minimizing a multivariable function
View Description Hide DescriptionThe method of minimization of a multivariable function based on the statistical mechanics analogy with a fictitious physical system of many particles is proposed. The function is assumed to be the Hamiltonian of the fictitious physical system to fit the global minimum of the function and the ground state ‘‘energy’’ of the fictitious system. In this model the global minimum search can be imitated by various relaxation processes in the fictitious system described by statistical mechanics. These relaxation processes lead to the equilibrium state, which is the ground state at the zero temperature limit. The imitation of a relaxation process confers to the minimization procedure the advantage of a relaxation process in a real physical system: because of thermal fluctuations a real system cannot be trapped by metastable states related to local minima. It always reaches the equilibrium state. The simulations of the relaxation processes based on the macroscopic kinetic equations and on the Monte Carlo algorithms are discussed. The new Monte Carlo algorithm based on the simulation of random walks of the representative point of the system in multidimensional phase space of the variables of the function under investigation is proposed. Unlike the conventional Metropolis–Rosenbluths–Tellers Monte Carlo method, each elementary transition in the proposed algorithm results in simultaneous movement of all atoms of the system, i.e., it generates a fluctuation involving any number of atoms.

R‐mer filling with general range‐R cooperative effects
View Description Hide DescriptionAn exact closed form solution is obtained for the time dependence of the coverage of a homogeneous, infinite, one‐dimensional lattice filled irreversibly and cooperatively by R‐mers. Cooperative effects, n o t assumed to be reflection invariant, may extend up to range R. Previously available exact solutions for random filling and nearest neighbor cooperative effects are recovered. For dimer filling with genuine range‐2 cooperative effects it is found that autoretardative and autocatalytic rate regimes may lead to the s a m e saturation coverage. Various adsorption schemes are considered.

On the phase transition of the three‐dimensional Percus–Yevick equation for an arbitrary potential of finite range
View Description Hide DescriptionA qualitative study of the three‐dimensional Percus–Yevick (PY) equation by means of Baxter’s relations is considered for an arbitrary potential of finite range l by a perturbation method. It is shown that the PY equation has a unique solution Y(r,η,β§) and a unique solution Q(r,η,β§) if the following conditions are satisfied: (i) 0<η<0.175, (ii) 0<β§<(β§)_{0}, (iii) Sup_{ r∈[0,l]}‖Q _{ n }‖<n! and Sup_{ r>0}‖Y _{ n }(r)‖<n!, where both Q and Y are continuous functions of the reduced density η, and can expressed as absolutely and uniformly convergent series Y=∑^{∞} _{ n=0}(1/n!)(β§)^{ n } Y _{ n }(r,η), Q=∑^{∞} _{ n=0}(1/n!)(β§)^{ n } Q _{ n }(r,η) within the radius of convergence of the inverse reduced temperature (β§)_{0}. As functions of r, Q∈C ^{(} ^{0} ^{)}[0,l] with Q(l)=0, whereas Y is continuous for r≥0 except for a possible finite discontinuity at r=1, and Y−r→0 exponentially as r→∞. Based on the solution of Y and Q, the isothermal compressibility K _{ T } =K T(∂ρ/∂P)_{ T } is a continuous and bounded function of η. As η→η_{ c } =0.175, K _{ T } becomes divergent. The critical density η_{ c } (or ρ_{ c }) is independent of the range of the attractive potential l. On the other hand, the critical temperature (β§)_{ c } is determined by the positive root of F(β§)=12η∫^{ l } _{0} Q(r)d r =1, which depends explicitly on the value of l.

An alternate proof of ultraviolet stability of the two‐dimensional massive sine–Gordon field theory in all regions of collapse
View Description Hide DescriptionThe upper bound for the ultraviolet stability of the two‐dimensional cosine interaction ∫_{Λ}:cos αφ_{ξ}:dξ, Λ⊆R^{2}, in finite volume Λ is proven for α^{2} ∈ [4π,8π[, where the theory has been shown to be superrenormalizable [see, e.g., G. Gallavotti, Rev. Mod. Phys. 5 7, 471 (1985)]. Ultraviolet stability in this interval was proven previously (F. Nicolò, J. Renn, and A.Steinmann, ‘‘On the massive sine–Gordon equation in all regions of collapse,’’ preprint II Università di Roma, 1985). Here we give a second proof using renormalization group methods based on a multiscale decomposition of the field by showing that the large fluctuations may be controlled by their small probability. The method essentially follows the one given by Nicolò [F. Nicolò, Commun. Math. Phys. 8 8, 681 (1983)] for α^{2} ∈ [4π, (32)/(5) π[.