An exactly solvable many-body problem in one dimension

For N impenetrable particles in one dimension where only the nearest and next-to-nearest neighbours interact, we obtain the complete spectrum both on a line and on a circle. Further, we establish a mapping between these N-body problems and the short-range Dyson model introduced recently to model intermediate spectral statistics in some systems using which we compute the two-point correlation function and prove the absence of long-range order in the corresponding many-body theory. Further, we also show the absence of off-diagonal long-range order in these systems.Comment: LaTeX, 4 pages, 1 figur

Off-diagonal long-range order in one-dimensional many-body problem

We prove that there is off-diagonal long-range order in the symmetrised version of the one-dimensional many-body problem presented by Jain and Khare (Phys. Lett. A262 (1999)35). This model is related to the short-range Dyson model employed to study intermediate statistics in systems like the Anderson model in three dimensions at the metal-insulator transition point and pseudointegrable billiards. To the best of our knowledge, this is the only example showing quantum phases and possibility of Bose-Einstein condensation in one-dimensional statistical mechanics.Comment: LaTeX2e, submitted to Phys. Lett.

Exact Solutions of the Two-Dimensional Discrete Nonlinear Schr\"odinger Equation with Saturable Nonlinearity

We show that the two-dimensional, nonlinear Schr\"odinger lattice with a saturable nonlinearity admits periodic and pulse-like exact solutions. We establish the general formalism for the stability considerations of these solutions and give examples of stability diagrams. Finally, we show that the effective Peierls-Nabarro barrier for the pulse-like soliton solution is zero

Affine Toda-Sutherland Systems

A cross between two well-known integrable multi-particle dynamics, an affine Toda molecule and a Sutherland system, is introduced for any affine root system. Though it is not completely integrable but partially integrable, or quasi exactly solvable, it inherits many remarkable properties from the parents. The equilibrium position is algebraic, i.e. proportional to the Weyl vector. The frequencies of small oscillations near equilibrium are proportional to the affine Toda masses, which are essential ingredients of the exact factorisable S-matrices of affine Toda field theories. Some lower lying frequencies are integer times a coupling constant for which the corresponding exact quantum eigenvalues and eigenfunctions are obtained. An affine Toda-Calogero system, with a corresponding rational potential, is also discussed.Comment: LaTeX2e 22 pages with amsfonts and graphicx, 5 eps figure

A class of N-body problems with nearest- and next-to-nearest neighbour interactions

We obtain the exact ground state and a part of the excitation spectrum in one dimension on a line and the exact ground state on a circle in a case where N particles are interacting via nearest- and next-to-nearest neighbour interactions. Further, using the exact ground-state, we establish a mapping between these N-body problems and the short-range Dyson models introduced recently to model intermediate spectral statistics. Using this mapping we compute the one- and two-point functions of a related many-body theory and show that there is no long-range order in the thermodynamic limit. However, quite remarkably, we prove the existence of an off-diagonal long-range order in the symmetrised version of the related many-body theory. Generalisation of the models to other root systems is also considered. Besides, we also generalize the model on the full line to higher dimensions. Finally, we consider a model in two dimensions in which all the states exhibit novel correlations.Comment: LaTeX2e, 40 pages, 2 figures, submitted to Nucl. Phys. B [FS

New Shape Invariant Potentials in Supersymmetric Quantum Mechanics

Quantum mechanical potentials satisfying the property of shape invariance are well known to be algebraically solvable. Using a scaling ansatz for the change of parameters, we obtain a large class of new shape invariant potentials which are reflectionless and possess an infinite number of bound states. They can be viewed as q-deformations of the single soliton solution corresponding to the Rosen-Morse potential. Explicit expressions for energy eigenvalues, eigenfunctions and transmission coefficients are given. Included in our potentials as a special case is the self-similar potential recently discussed by Shabat and Spiridonov.Comment: 8pages, Te

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