Excitation of hydrogen molecule by electron impact, III - Singlet-triplet excitations

Exchange excitation of hydrogen molecule by electron impact from ground to triplet electronic stat

Local Identities Involving Jacobi Elliptic Functions

We derive a number of local identities of arbitrary rank involving Jacobi elliptic functions and use them to obtain several new results. First, we present an alternative, simpler derivation of the cyclic identities discovered by us recently, along with an extension to several new cyclic identities of arbitrary rank. Second, we obtain a generalization to cyclic identities in which successive terms have a multiplicative phase factor exp(2i\pi/s), where s is any integer. Third, we systematize the local identities by deriving four local ``master identities'' analogous to the master identities for the cyclic sums discussed by us previously. Fourth, we point out that many of the local identities can be thought of as exact discretizations of standard nonlinear differential equations satisfied by the Jacobian elliptic functions. Finally, we obtain explicit answers for a number of definite integrals and simpler forms for several indefinite integrals involving Jacobi elliptic functions.Comment: 47 page

Chemical Enrichment at High Redshifts

We have tried to understand the recent observations related to metallicity in Ly $\alpha$ forest clouds in the framework of the two component model suggested by Chiba & Nath (1997). We find that even if the mini-halos were chemically enriched by an earlier generation of stars, to have [C/H] $\simeq$ -2.5, the number of C IV lines with column density $>10^{12} cm^{-2}$, contributed by the mini-halos, at the redshift of 3, would be only about 10% of the total number of lines, for a chemical enrichment rate of $(1+z)^{-3}$ in the galaxies. Recently reported absence of heavy element lines associated with most of the Ly $\alpha$ lines with H I column density between $10^{13.5} cm^{-2}$ and $10^{14} cm^{-2}$ by Lu et al (1998), if correct, gives an upper limit on [C/H]=-3.7, not only in the mini-halos, but also in the outer parts of galactic halos. This is consistent with the results of numerical simulations, according to which, the chemical elements associated with the Ly $\alpha$ clouds are formed in situ in clouds, rather than in an earlier generation of stars. However, the mean value of $7 \times 10^{-3}$ for the column density ratio of C IV and H I, determined by Cowie and Songaila (1998) for low Lyman alpha optical depths, implies an abundance of [C/H] =-2.5 in mini-halos as well as in most of the region in galactic halos, presumably enriched by an earlier generation of stars. The redshift and column density distribution of C IV has been shown to be in reasonable agreement with the observations.Comment: 23 pages, 6 figures, To appear in Astrophysical Journa

Truncated Harmonic Osillator and Parasupersymmetric Quantum Mechanics

We discuss in detail the parasupersymmetric quantum mechanics of arbitrary order where the parasupersymmetry is between the normal bosons and those corresponding to the truncated harmonic oscillator. We show that even though the parasusy algebra is different from that of the usual parasusy quantum mechanics, still the consequences of the two are identical. We further show that the parasupersymmetric quantum mechanics of arbitrary order p can also be rewritten in terms of p supercharges (i.e. all of which obey $Q_i^{2} = 0$). However, the Hamiltonian cannot be expressed in a simple form in terms of the p supercharges except in a special case. A model of conformal parasupersymmetry is also discussed and it is shown that in this case, the p supercharges, the p conformal supercharges along with Hamiltonian H, conformal generator K and dilatation generator D form a closed algebra.Comment: 9 page

Linear Superposition in Nonlinear Equations

Even though the KdV and modified KdV equations are nonlinear, we show that suitable linear combinations of known periodic solutions involving Jacobi elliptic functions yield a large class of additional solutions. This procedure works by virtue of some remarkable new identities satisfied by the elliptic functions.Comment: 7 pages, 1 figur

Soliton Lattice and Single Soliton Solutions of the Associated Lam\'e and Lam\'e Potentials

We obtain the exact nontopological soliton lattice solutions of the Associated Lam\'e equation in different parameter regimes and compute the corresponding energy for each of these solutions. We show that in specific limits these solutions give rise to nontopological (pulse-like) single solitons, as well as to different types of topological (kink-like) single soliton solutions of the Associated Lam\'e equation. Following Manton, we also compute, as an illustration, the asymptotic interaction energy between these soliton solutions in one particular case. Finally, in specific limits, we deduce the soliton lattices, as well as the topological single soliton solutions of the Lam\'e equation, and also the sine-Gordon soliton solution.Comment: 23 pages, 5 figures. Submitted to J. Math. Phy

Periodic Solutions of Nonlinear Equations Obtained by Linear Superposition

We show that a type of linear superposition principle works for several nonlinear differential equations. Using this approach, we find periodic solutions of the Kadomtsev-Petviashvili (KP) equation, the nonlinear Schrodinger (NLS) equation, the $\lambda \phi^4$ model, the sine-Gordon equation and the Boussinesq equation by making appropriate linear superpositions of known periodic solutions. This unusual procedure for generating solutions is successful as a consequence of some powerful, recently discovered, cyclic identities satisfied by the Jacobi elliptic functions.Comment: 19 pages, 4 figure

Exceptional orthogonal polynomials, QHJ formalism and SWKB quantization condition

We study the quantum Hamilton-Jacobi (QHJ) equation of the recently obtained exactly solvable models, related to the newly discovered exceptional polynomials and show that the QHJ formalism reproduces the exact eigenvalues and the eigenfunctions. The fact that the eigenfunctions have zeros and poles in complex locations leads to an unconventional singularity structure of the quantum momentum function $p(x)$, the logarithmic derivative of the wave function, which forms the crux of the QHJ approach to quantization. A comparison of the singularity structure for these systems with the known exactly solvable and quasi-exactly solvable models reveals interesting differences. We find that the singularities of the momentum function for these new potentials lie between the above two distinct models, sharing similarities with both of them. This prompted us to examine the exactness of the supersymmetric WKB (SWKB) quantization condition. The interesting singularity structure of $p(x)$ and of the superpotential for these models has important consequences for the SWKB rule and in our proof of its exactness for these quantal systems.Comment: 10 pages with 1 table,i figure. Errors rectified, manuscript rewritten, new references adde