375 research outputs found

### 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

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