For N impenetrable particles in one dimension where only the nearest and the 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 an off-diagonal long-range order in the system.

A PVC-based membrane of 2,2,2-cryptand exhibits a very good response for Zn(2+) in a wide concentration range (from 2.06 ppm to 6.54 × 10(3) ppm) with a slope of 22.0 mV/decade of Zn(2+) concentration. The response time of the sensor is <10 s, and the membrane can be used for more than 3 months without any observed divergence in potentials. The proposed sensor exhibits very good selectivity for Zn(2+) over other cations and can be used in a wide pH range (2.8-7.0). It has also been possible to use this assembly as an indicator electrode in potentiometric titrations involving zinc ions.

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Poly(viny1 chloride) (PVC)-based membrane of pentathia-15-crown-5 exhibits good potentiometric response for Hg2+ over a wide concentration range (2.51 x to 1.00 x lo-' mol d M 3 ) with a slope of 32.1 mV per decade of Hg2+ concentration. The response time of the sensor is as fast as 20 s. The electrode has been used for a period of six weeks and exhibits fairly good discriminating ability towards Hg2+ in comparison to alkali, alkaline and some heavy metal ions. The electrode can be used in the pH range from 2.7 to 5.0.

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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 the case where N particles are interacting via nearest and nextto-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 oneand two-point functions of a related many-body theory and show the absence of long-range order in the thermodynamic limit. 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.

The m. /3-rhombus billiard is an example of the simplest pseudointegrable system having an invariant integral surface of genus g =2. We examine the fluctuation properties of eigenvalue sequences belonging to the "pure rhombus" modes {the eigenfunctions take nonzero values on the shorter diagonal). The nearest-neighbor spacing statistics follow the Berry-Robnik distribution with a chaotic fraction v (corresponding to the Liouville measure of the chaotic subspace) equal to 0.8. The spectral rigidity closely agrees with such a partitioning of phase space. The nodal patterns and the path correlation function exhibit irregularity for most of the corresponding eigenfunctions.Though the amplitude distributions for these closely approximate a Gaussian distribution, the spatial correlations do not agree well with the well-known Bessel oscillations. A few eigenfunctions, however, show regularity. These are localized in those regions of configuration space where the bouncing-ball modes form rectangular bands. The Born-Oppenheimer approximation offers a suitable explanation in terms of a confining potential, and the agreement between the exact and adiabatic eigenvalues improve at higher energies. On the basis of these observations, it turns out that the quantities v and v are, in fact, the fractions of regular and irregular states in the eigenvalue sequence under consideration. Thus irregular eigenfunctions do occur for systems with zero Kolmogorov entropy, and the eigenvalue sequence corresponding to these yield Gaussian orthogonal ensemble statistics.