TheCP N−1 1/N-action by inverse scattering transformation in angular momentum

The effective action which generates 1/N expansion of theCP _N–1 model in two dimensions is studied here by inverse-problem methods. The action contains a functional determinant, in which auxiliary scalar and vector fields are assumed to have a spherical symmetry. This leads to the introduction, as an associated linear problem, of a radial Schrödinger equation with two potentialsv and , and a potential-dependent centrifugal term {(–r)²/r ²–1/4r ²}. The full inverse scattering formalism is developed here for this diffusion problem. It is formulated in terms of two-component Jost solutions, and leads to a matricial Gel'fand-Levitan-Marchenko equation. The scattering data associated to the potentials by this IST are then used to obtain a closed local form for the whole effective action. This is indeed possible for theCP _N–1 model, owing to the classical integrability. Moreover it is found that no spherically symmetric instanton exists in this case. However the absence of supplementary informations on the 1/N series, due to the non-integrability at quantum level, does not allow safe quantitative conclusions on the general behaviour of the 1/N series at large orders.Laboratoire associé au CNRS UA 280     

LETTER: A Simple Quantum Cosmology

A simple and surprisingly realistic model of the origin of the universe can be developed using the Friedmann equation from general relativity, elementary quantum mechanics, and the experimental values of , c, G and the proton mass m _p. The model assumes there are N space dimensions (with N > 6), and the potential constraining the radius r of the invisible N – 3 compact dimensions varies as r ⁴. In this model, the universe has zero total energy and is created from nothing. There is no initial singularity. If space-time is eleven dimensional, as required by M theory, the scalar field corresponding to the size of the compact dimensions inflates the universe by about 26 orders of magnitude (60 e-folds). If H ₀ = 65 km sec^–1 Mpc^–1, the energy density of the scalar field after inflation results in = 0.68, in agreement with recent COBE and Type SNe Ia supernova data.     

Exact Solutions of the Schrödinger Equation with Inverse-Power Potential

The Schrödinger equation for stationary states is studied in a central potential V(r) proportional to r ^– in an arbitrary number of spatial dimensions. The presence of a single term in the potential makes it impossible to use previous algorithms, which only work for quasi-exactly-solvable problems. Nevertheless, the analysis of the stationary Schrödinger equation in the neighbourhood of the origin and of the point at infinity is found to provide relevant information about the desired solutions for all values of the radial coordinate. The original eigenvalue equation is mapped into a differential equation with milder singularities, and the role played by the particular case = 4 is elucidated. In general, whenever the parameter is even and larger than 4, a recursive algorithm for the evaluation of eigenfunctions is obtained. Eventually, in the particular case of two spatial dimensions, the exact form of the ground-state wave function is obtained for a potential containing a finite number of inverse powers of r, with the associated energy eigenvalue.     

Exactly Complete Solutions of the Pseudoharmonic Potential in N-Dimensions

We present analytically the exact solutions of the Schrödinger equation in the N-dimensional spaces for the pseudoharmonic oscillator potential by means of the ansatz method. The energy eigenvalues of the bound states are easily calculated from this eigenfunction ansatz. The normalized wavefunctions are also obtained. A realization of the ladder operators for the wavefunctions is studied and we deduced that these operators satisfy the commutation relations of the generators of the dynamical group SU(1,1). Some expectation values for 〈r ^?2〉, 〈r ²〉, 〈T〉, 〈V〉, 〈H〉, 〈p ²〉 and the virial theorem for the pseudoharmonic oscillator potential in an arbitrary number of dimensions are obtained by means of the Hellmann–Feynman theorems. Each solution obtained is dimensions and parameters dependent.     

Quantum Spectrum of Some Anharmonic Central Potentials: Wave Functions Ansatz

Based on the ansatz to the wave functions, the quasi-exact solutions of the 2D Schrödinger equation with some anharmonic potentials are reviewed and analyzed if admitting restrictions on the parameters of the potential and the angular momentum m. These potentials are taken as the screened Coulomb potential V(r)=a/r+b/(r+), the singular one-fraction power one V(r)=ar ^–1/2+br ^–3/2 and the singular two-fraction one V(r)=ar ^2/3+br ^–2/3+cr ^–4/3. The latter one is found that the hidden symmetry exists if substituting r–ir. It will reverse the signs of E and c of quantum system, leaving the remaining parameters invariant.     

A New Approach to the Relativistic Schrödinger Equation with Central Potential: Ansatz Method

Applying an ansatz to the eigenfunction, we obtain the exact closed-form solutions of the relativistic Schrödinger equation with the potential V(r) = –a/r + b/r^1/2 both in three dimensions and in two dimensions. The restrictions on the parameters of the given potential and the angular momentum quantum number are also presented.     

Exact decay of correlations for mixtures and rotators

We give the exact asymptotic form, at low activity, of the correlations of a classical fluid consisting of several species of particles interacting by means of integrable two-body potentials. Our results also extend to classical dipoles withr ^–3 potential in two dimensions.     

The inverse scattering transformation in the angular momentum plane

The inverse scattering transformation (IST) with the angular momentum () as a spectral variable turned out to be a useful method to deal with rotationally invariant problems in field theory at higher spatial dimensions [Refs. (1) and (2)]. We derive the direct and inverse scattering problems for thev-dimensional radial Schrödinger equation for variable and fixed energy (negative or positive). We determine the scattering data (SD) in one to one correspondence with the potential and derive the corresponding Gelfand-Levitan equation. A family of exactly solvable potentials for any and fixed energy is obtained. The trace identities associated to this IST are derived for both signs of the energy. They relate integrals of local polynomials in the potential and its derivatives timesr ^2n–1(n1) with the SD. The presence of this power ofr makes these relations useful in higher dimensions.Laboratoire associé au CNRS     

Exact Solutions of the Two-Dimensional Schrödinger Equation with Certain Central Potentials

By applying an ansatz to the eigenfunction, an exact closed-form solution of theSchrödinger equation in two dimension is obtained with the potentials V(r) =ar ² + br ⁴ + cr ⁶,V(r) = ar + br² + cr ^–1,and V(r) = ar ² + br ^–2+ cr ^–4 + dr ^–6,respectively. The restrictions on the parameters of the given potential andthe angular momentum m are obtained.     

Large orders in the 1/N perturbation theory by inverse scattering in one dimension

When one tries to compute large orders in the 1/N series à la Lipatov a complicated non-linear equation for the instanton is found in ø⁴ or non-linear sigma models.We solve here this equation in the one-dimensional case (quantum mechanics) by inverse scattering techniques. From the instanton solutions we obtain theK ^th order of the 1/N perturbation theory up to 0(K ^–1) for the 0(N) symmetric anharmonic oscillator and up to a factor 0(K ⁰) for a non-symmetric model. In the symmetric case we agree with results recently obtained in quantum mechanics by Hikami and Brézin following a different procedure. For the non-symmetric anharmonic oscillator we believe our formulae are new.     

New proofs of the existence of the Feigenbaum functions

A new proof of the existence of analytic, unimodal solutions of the Cvitanovi-Feigenbaum functional equation g(x) = –g(g(–x)),g(x) 1 - const|x|^r at 0, valid for all in (0, 1), is given, and the existence of the Eckmann-Wittwer functions [8] is recovered. The method also provides the existence of solutions for certain given values ofr, and in particular, forr=2, a proof requiring no computer.     

Overlap contribution to57Fe3+ isomer shifts in fluorides and oxides

We have calculated the overlap contribution to the contact charge density at the⁵⁷Fe³⁺ site in fluoride and oxide lattices, FeF₃, CaFe₂O₄, CaBaFe₄O₈ and KFeO₂. The trend in isomer shifts in going from one lattice to another is explained in terms of the overlap effects. The experimental values of the change in isomer shift from lattice to lattice is used to obtain the change in nuclear size. The values of r/r _N are found to lie between –0.81×10^–4 and –1.56×10^–4.     

Exact Analytical Solution of the Schrödinger Equation for an N-Identical Body-Force System

A mathematical method is presented for solving the Schr?dinger equation for a system of identical body forces. The N-body forces are more easily introduced and treated within the hyperspherical harmonics. The problem of the N-body potential has been used at the level of both classical and quantum mechanics. The hypercentral interacting potential is assumed to depend on the hyperradius x = (ξ₁² + ξ₂² + ⋯ + ξ_N−1²)^1/2 only, where ξ₁,ξ₂,…,ξ_N−1 are Jacobi relative coordinates which are functions of N-particle relative positions r₁₂,r₂₃,…,r_N1. The problem of the harmonic oscillator and the Coulomb-type potential has been widely studied in different contexts. Using the N-body potential V(x) = ax² + bx − (c/x) as an example, and assuming an ansatz for the eigenfunction, an exact analytical solution of the Schr?dinger equation for an N-body system in three dimensions is obtained. This method is also applicable to some other types of potentials for N-identical interacting particles.     

A necessary and sufficient condition for the existence of multisolitons in a self-dual gauged sigma model

This paper presents a resolution of the gaugedO(3) sigma model proposed by B.J. Schroers in which the matter field ø mapsR ² intoS ² while the vector gauge potential gives rise to a magnetic field. It is shown that for each natural numberN there are solutions to saturate the classical energy lower boundE4N for the field configurations in the topological family deg(ø)=N if and only ifN1. Furthermore the solutions obtained depend on at least 4N–3 continuous parameters, the associated magnetic flux can assume its value in an open interval, and the decay rates of the field strengths may be specified in a suitable range. These solutions are multisolitons represented byN prescribed lumps of the magnetic field, simulatingN identical particles in equilibrium, and are governed by a nonlinear elliptic equation with both vortex and anti-vortex source terms.Research supported in part by NSF under grant DMS-9400243 (DMS-9596041).     

On existence and behaviour of asymptotically flat solutions to the stationary Einstein equations

We show existence and uniqueness of asymptotically flat solutions to the stationary Einstein equations inS=³–B _r , whereB _r is a ball of radiousr>0, when a small enough continuous complex function û on S is given. Regularity and decay estimates imply that these solutions are analytic in the interior ofS and also at infinity, when suitably conformally rescaled.     

Transport in a disordered medium: Analysis and Monte Carlo simulation

We consider a classical stochastic model describing particle transport on a lattice with randomly distributed nearest-neighbor transition rates. Applying an effective medium theory to the model, we determine average properties related to the particle's dynamics ind-dimensions. In particular, we calculate the mean-square displacement, and the fourth moment of the displacement in one-, two- and three dimensions. The results compare favorably with Monte Carlo simulations of the model. We also present preliminary results for the velocity autocorrelation function.An aspect of the bond percolation problem, which is a special case of the stochastic model is investigated; the average inverse cluster size, <N _c ^–1>, is calculated. In one dimension the expression for this quantity is exact and in higher dimensions our results are very accurate not too close to the percolation concentration.     

Investigation of the stability of the inverse spectroscopic problem of diatomic molecules

The stability and convergence of the inverse spectroscopic problem is investigated for diatomic molecules in the¹ electron state for a potential representation by series in the Dunham variable z_d=(r-r_e)/r_e, the Ogilvie-Tipping z_ot=(r-r_e)/(r + r_e), the Simmond-Parr-Finlan z_S= (r-r_e)/r and the Tucker z_t=sign (p)[1 -(r_e/r)^P]. None among the representations under investigation was that which would give a significant systematic improvement in stability as compared with others. The convergence is related to the selection of the initial approximation in the method of least squares, and is practically independent of the functional form of the potential. For the Dunham representation the solutions of the inverse problem yield quantities close to the true values of the expansion coefficients.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 98–101, May, 1984.     

Nearest neighbor effect on capture cross sections measured in IR phototransients on Si:In

The extrinsic photoconductive decay at T=20–100 K is analyzed in FZ-grown Si: In material after pulsed irradiation by a PbSSe infrared laser (=4 m). Trapping time constants (=10 ns-100 s) are resolved for the prevalent In acceptor (N _In=10¹⁶–10¹⁷ cm^–3) and for additional shallow acceptors B, Al, and the X(In)-center present at low concentrations (N=10¹²–10¹⁴ cm^–3). Hole capture cross sections determined for the acceptor levels show a large scatter over up to 4 orders of magnitude. It is shown that the capture cross section is dependent on all the dopant concentrations present in the sample due to nearest neighbor interaction. Due to the formation of donor-acceptor dipoles, the capture cross section assumes low values. A model calculation of the interaction based on only fundamental parameters of Si is in accordance with the experimental data within the experimental error. The hole capture cross sections for isolated acceptors are _p=1×10^–12, 1×10^–14, 1×10^–13, 2.5×10^–13 cm² for indium, X-center, aluminum, and boron at the temperatures T=95 K, 100 K, 70 K, 45 K, respectively.     

Differential Invariants of Immersions of Manifolds with Metric Fields

The problem of finding all r^th order differential invariants of immersions of manifolds with metric fields, with values in a left (G¹_m×G¹_n)-manifold is formulated. For obtaining the basis of higher order differential invariants the orbit reduction method is used. As a new result it appears that r^th order differential invariants depending on an immersion f:M N of smooth manifolds M and N and metric fields on them can be factorized through metrics, curvature tensors and their covariant differentials up to the order (r–2), and covariant differentials of the tangent mapping Tf up to the order r. The concept of a covariant differential of Tf is also introduced in this paper. The obtained results are geometrically interpreted as well.This research is supported by grants GAR 201/03/0512 and MSM 143100006.