Sudhir R. Jain

An exactly solvable many-body problem in one dimension and the short-range Dyson model

Abstract For N impenetrable particles in one dimension with upto next-to-nearest neighbours interaction, we obtain the exact ground state.We establish a mapping between these N-body problems and the short-range Dyson model introduced recently to model intermediate spectral statistics. We prove the absence of long-range order and off-diagonal long-range order in the corresponding many-body theory.

Time delay plots of unflavoured baryons

Abstract We explore the usefulness of the existing relations between the S -matrix and time delay in characterizing baryon resonances in pion–nucleon scattering. We draw attention to the fact that the existence of a positive maximum in time delay is a necessary criterion for the existence of a resonance and should be used as a constraint in conventional analyses which locate resonances from poles of the S -matrix and Argand diagrams. The usefulness of the time delay plots of resonances is demonstrated through a detailed analysis of the time delay in several partial waves of πN elastic scattering.

Pseudounitary symmetry and the Gaussian pseudounitary ensemble of random matrices

Employing the currently discussed notion of pseudo-Hermiticity, we define a pseudounitary group. Further, we develop a random matrix theory that is invariant under such a group and call this ensemble of pseudo-Hermitian random matrices the pseudounitary ensemble. We obtain exact results for the nearest-neighbor level-spacing distribution for (2 x 2) PT-invariant Hamiltonian matrices that have forms, approximately Sln(1/S) near zero spacing for three independent elements and approximately S for four independent elements. This shows a level repulsion in a marked distinction with an algebraic form S(beta) in the Wigner surmise. We believe that this paves the way for a description of varied phenomena in two-dimensional statistical mechanics, quantum chromodynamics, and so on.

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

We obtain the exact ground state and part of the excitation spectrum in one dimension on a line and the exact ground state on a circle in the case where the N particles are interacting via nearest- and next-to-nearest-neighbour interactions. Furthermore, 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 in the thermodynamic limit and show the absence of long-range order. However, quite remarkably, we prove the existence of an off-diagonal long-range order in the symmetrized version of the related many-body theory. Generalization 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.

Spacing distributions for rhombus billiards

We show that the spacing distributions of rational rhombus billiards fall in a family of universality classes distinctly different from the Wigner - Dyson family of random matrix theory and the Poisson distribution. Some of the distributions find explanation in a recent work of Bogomolny, Gerland, and Schmit. For the irrational billiards, despite ergodicity, we get the same distribution for the examples considered - once again, distinct from the Wigner - Dyson distributions. All the results are obtained numerically by a method that allows us to reach very high energies.

Gaussian ensemble of 2 × 2 pseudo-Hermitian random matrices

We present a random matrix theory for systems invariant under the joint action of parity, , and time reversal, , and, more generally, for pseudo-Hermitian systems. This brings out the appearance of the metric in a systematic way so that consistency with the postulates of quantum mechanics is maintained. Here we specialize only to 2 × 2 matrices and we construct a pseudo-unitary group. With explicit examples, nearest-neighbour level-spacing distributions for various classes of ensembles are found to exhibit a degree of level repulsion different from those hitherto known. This work is not only relevant to quantum chaos, but also to two-dimensional statistical mechanics and consistent non-local relativistic theories.

From random matrix theory to statistical mechanics - anyon gas

Abstract Motivated by numerical experiments and studies of quantum systems which are classically chaotic, we take the random matrix description of a hard-sphere gas to the statistical mechanical description. We apply this to the anyon gas and obtain a formal expression for the momentum distribution. Various limiting situations are discussed and are found to be in agreement with the well-known results on hard-sphere gas in the low-density regime.

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

We prove the existence of an off-diagonal long-range order in a one-dimensional many-body problem.

Nonlinear dynamics of high-j cranking model: A semi-classical approach

Abstract We present a complete discussion of the dynamical scenario supported by the cranking model for high- j orbital where we consider the effects of the nonlinear terms. We carry out a semiclassical quantization about the various fixed points of the model and obtain the alignment of particle angular momentum with the rotation axis under various conditions. Further, we establish the occurrence of second-order phase transition accompanied by a change of sign of aligned angular momentum. Possible applications of the model to deformed and superdeformed nuclei are considered.

Non-universal spectral rigidity of quantum pseudo-integrable billiards

We obtain the Dyson - Mehta -statistic for pseudo-integrable billiards and show that it is non-universal with a universal trend, also that this trend is similar to the one for integrable billiards. We present a formula, based on exact semiclassical calculations and the proliferation law of periodic orbits, which gives rigidity for the entire range of L. To consolidate our theory, we discuss several examples finding complete agreement with the numerical results, and also the underlying fundamental reasons for the non-universality.