Exact Ground State of Several N-body Problems With an N-body Potential

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 $-e^2/\sqrt{\sum_{i<j} (r_i - r_j)^2}$). Based on these and previously known examples, I conjecture that whenever an N-body problem is solvable in case the N-bodies are interacting via an oscillator potential, the same problem is also solvable in case they are interacting via the N-body potential. Based on several examples, I also conjecture that in either case one can always add an N-body potential of the form $\beta^2/{\sum_{i} r_i^2}$ and the problem is still solvable except that the degeneracy in the bound state spectrum is now much reduced.Comment: 32 pages, No figur

Chern-Simons Term and Charged Vortices in Abelian and Nonabelian Gauge Theories

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.Comment: To be published in the Proceedings of Indian National Science Academy, Part A-physical Science

Supersymmetry in Quantum Mechanics

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.Comment: Latex file, 4 figures, Lecture Notes presented at EALF 2004, Will be published in the AIP Conference Proceedings AI

Some Exact Results for Mid-Band and Zero Band-Gap States of Associated Lame Potentials

Applying certain known theorems about one-dimensional periodic potentials, we show that the energy spectrum of the associated Lam\'{e} potentials $$a(a+1)m~{\rm sn}^2(x,m)+b(b+1)m~{\rm cn}^2(x,m)/{\rm dn}^2(x,m)$$ consists of a finite number of bound bands followed by a continuum band when both $a$ and $b$ take integer values. Further, if $a$ and $b$ are unequal integers, we show that there must exist some zero band-gap states, i.e. doubly degenerate states with the same number of nodes. More generally, in case $a$ and $b$ are not integers, but either $a + b$ or $a - b$ is an integer ($a \ne b$), we again show that several of the band-gaps vanish due to degeneracy of states with the same number of nodes. Finally, when either $a$ or $b$ is an integer and the other takes a half-integral value, we obtain exact analytic solutions for several mid-band states.Comment: 18 pages, 2 figure

Cyclic Identities Involving Jacobi Elliptic Functions

We state and discuss numerous mathematical identities involving Jacobi elliptic functions sn(x,m), cn(x,m), dn(x,m), where m is the elliptic modulus parameter. In all identities, the arguments of the Jacobi functions are separated by either 2K(m)/p or 4K(m)/p, where p is an integer and K(m) is the complete elliptic integral of the first kind. Each p-point identity of rank r involves a cyclic homogeneous polynomial of degree r (in Jacobi elliptic functions with p equally spaced arguments) related to other cyclic homogeneous polynomials of degree r-2 or smaller. Identities corresponding to small values of p,r are readily established algebraically using standard properties of Jacobi elliptic functions, whereas identities with higher values of p,r are easily verified numerically using advanced mathematical software packages.Comment: 14 pages, 0 figure

Classical and quantum mechanics of a particle on a rotating loop

The toy model of a particle on a vertical rotating circle in the presence of uniform gravitational/ magnetic fields is explored in detail. After an analysis of the classical mechanics of the problem we then discuss the quantum mechanics from both exact and semi--classical standpoints. Exact solutions of the Schrodinger equation are obtained in some cases by diverse methods. Instantons, bounces are constructed and semi-classical, leading order tunneling amplitudes/decay rates are written down. We also investigate qualitatively the nature of small oscillations about the kink/bounce solutions. Finally, the connections of these toy examples with field theoretic and statistical mechanical models of relevance are pointed out.Comment: 30 pages, RevTex, 7 figure

Superposition of Elliptic Functions as Solutions For a Large Number of Nonlinear Equations

For a large number of nonlinear equations, both discrete and continuum, we demonstrate a kind of linear superposition. We show that whenever a nonlinear equation admits solutions in terms of both Jacobi elliptic functions \cn(x,m) and \dn(x,m) with modulus $m$, then it also admits solutions in terms of their sum as well as difference. We have checked this in the case of several nonlinear equations such as the nonlinear Schr\"odinger equation, MKdV, a mixed KdV-MKdV system, a mixed quadratic-cubic nonlinear Schr\"odinger equation, the Ablowitz-Ladik equation, the saturable nonlinear Schr\"odinger equation, $\lambda \phi^4$, the discrete MKdV as well as for several coupled field equations. Further, for a large number of nonlinear equations, we show that whenever a nonlinear equation admits a periodic solution in terms of \dn^2(x,m), it also admits solutions in terms of \dn^2(x,m) \pm \sqrt{m} \cn(x,m) \dn(x,m), even though \cn(x,m) \dn(x,m) is not a solution of these nonlinear equations. Finally, we also obtain superposed solutions of various forms for several coupled nonlinear equations.Comment: 40 pages, no figure