Abstract:
I obtain the exact ground state of $N$-fermions in $D$-dimensions $(D \geq 2)$ in case the $N$ particles are interacting via long-ranged two-body and three-body interactions and further they are also interacting via the harmonic oscillator potential. I also obtain the $N$-fermion ground state in case the oscillator potential is replaced by an $N$-body Coulomb-like interaction.

Abstract:
On the occasion of the birth centenary of the discovery of electron, I discuss its role in the area of Elementary Particle Physics. I emphasize that the discovery of electron marks the end of the speculation era lasting more than 2500 years. The key developments leading to the discovery of electron by J.J. Thomson are mentioned. The standard model is briefly described. It is emphasized that this model is unsatisfactory in several respects, however no further progress beyond standard model is possible unless there is a dramatic advance in instrumentation.

Abstract:
A class of exact solutions are obtained for the problem of N-anyons interacting via the N-body potential $V (\vec x_1,\vec x_2,...,\vec x_N)$ = $-{e^2\over\sqrt{{1\over N}\sum_{i

Abstract:
In this article we review some of the recent advances regarding the charged vortex solutions in abelian and nonabelian gauge theories with Chern-Simons (CS) term in two space dimensions. Since these nontrivial results are essentially because of the CS term, hence, we first discuss in some detail the various properties of the CS term in two space dimensions. In particular, it is pointed out that this parity (P) and time reversal (T) violating but gauge invariant term when added to the Maxwell Lagrangian gives a massive gauge quanta and yet the theory is still gauge invariant. Further, the vacuum of such a theory shows the magneto-electric effect. Besides, we show that the CS term can also be generated by spontaneous symmetry breaking as well as by radiative corrections. A detailed discussion about Coleman-Hill theorem is also given which aserts that the parity-odd piece of the vacuum polarization tensor at zero momentum transfer is unaffected by two and multi-loop effects. Topological quantization of the coefficient of the CS term in nonabelian gauge theories is also elaborated in some detail. One of the dramatic effect of the CS term is that the vortices of the abelian (as well as nonabelian) Higgs model now acquire finite quantized charge and angular momentum. The various properties of these vortices are discussed at length with special emphasis on some of the recent developments including the discovery of the self-dual charged vortex solutions.

Abstract:
Complete energy spectrum is obtained for the quantum mechanical problem of N one dimensional equal mass particles interacting via potential $$V(x_1,x_2,...,x_N) = g\sum^N_{i < j}{1\over (x_i-x_j)^2} - {\alpha\over \sqrt{\sum_{i < j} (x_i-x_j)^2}}$$ Further, it is shown that scattering configuration, characterized by initial momenta $p_i (i=1,2,...,N)$ goes over into a final configuration characterized uniquely by the final momenta $p'_i$ with $p'_i=p_{N+1-i}$.

Abstract:
I show that the potential $$V(x,m) = \big [\frac{b^2}{4}-m(1-m)a(a+1) \big ]\frac{\sn^2 (x,m)}{\dn^2 (x,m)} -b(a+{1/2}) \frac{\cn (x,m)}{\dn^2 (x,m)}$$ constitutes a QES band-structure problem in one dimension. In particular, I show that for any positive integral or half-integral $a$, $2a+1$ band edge eigenvalues and eigenfunctions can be obtained analytically. In the limit of m going to 0 or 1, I recover the well known results for the QES double sine-Gordon or double sinh-Gordon equations respectively. As a by product, I also obtain the boundstate eigenvalues and eigenfunctions of the potential $$V(x) = \big [\frac{\beta^2}{4}-a(a+1) \big ] \sech^2 x +\beta(a+{1/2})\sech x\tanh x$$ in case $a$ is any positive integer or half-integer.

Abstract:
I consider several N-body problems for which exact (bosonic) ground state and a class of excited states are known in case the N-bodies are also interacting via harmonic oscillator potential. I show that for all these problems the exact (bosonic) ground state and a class of excited states can also be obtained in case they interact via an N-body potential of the form $-e^2/\sqrt{\sumr^2_i}$ (or $-e^2/\sqrt{\sum_{i

Abstract:
Exactly twenty five years ago the world of high energy physics was set on fire by the discovery of a new particle with an unusually narrow width at 3095 MeV, known popularly as the $J/\Psi$ revolution. This discovery was very decisive in our understanding as well as formulating the current picture regarding the basic constituents of nature. I look back at the discovery, pointing out how unexpected, dramatic and significant it was.

Abstract:
An elementary introduction is given to the subject of Supersymmetry in Quantum Mechanics. We demonstrate with explicit examples that given a solvable problem in quantum mechanics with n bound states, one can construct new exactly solvable n Hamiltonians having n-1,n-2,...,0 bound states. The relationship between the eigenvalues, eigenfunctions and scattering matrix of the supersymmetric partner potentials is derived and a class of reflectionless potentials are explicitly constructed. We extend the operator method of solving the one-dimensional harmonic oscillator problem to a class of potentials called shape invariant potentials. Further, we show that given any potential with at least one bound state, one can very easily construct one continuous parameter family of potentials having same eigenvalues and s-matrix. The supersymmetry inspired WKB approximation (SWKB) is also discussed and it is shown that unlike the usual WKB, the lowest order SWKB approximation is exact for the shape invariant potentials. Finally, we also construct new exactly solvable periodic potentials by using the machinery of supersymmetric quantum mechanics.

Abstract:
The question of anyons and fractional statistics in field theories in 2+1 dimensions with Chern-Simons (CS) term is discussed in some detail. Arguments are spelled out as to why fractional statistics is only possible in two space dimensions. This phenomenon is most naturally discussed within the framework of field theories with CS term, hence as a prelude to this discussion I first discuss the various properties of the CS term. In particular its role as a gauge field mass term is emphasized. In the presence of the CS term, anyons can appear in two different ways i.e. either as soliton of the corresponding field theory or as a fundamental quanta carrying fractional statistics and both approaches are elaborated in some detail.