Tables of integrals

Lie Theory and Special Functions

Coherent states are derived for one-dimensional systems generated by supersymmetry from an initial Hamiltonian with a purely discrete spectrum for which the levels depend analytically on their subindex. It is shown that the algebra of the initial system is inherited by its SUSY partners in the subspace associated to the isospectral part or the spectrum. The technique is applied to the harmonic oscillator, infinite well and trigonometric Poeschl-Teller potentials.

Coherent states for Hamiltonians generated by supersymmetry

Coherent states are derived for one-dimensional systems generated by supersymmetry from an initial Hamiltonian with a purely discrete spectrum for which the levels depend analytically on their subindex. It is shown that the algebra of the initial system is inherited by its SUSY partners in the subspace associated with the isospectral part or the spectrum. The technique is applied to the harmonic oscillator, infinite well and trigonometric Poschl–Teller potentials.

http://iopscience.iop.org/article/10.1088/1751-8113/40/24/015/pdf

The position and momentum space information entropies, of the ground state of the Poeschl-Teller potential, are exactly evaluated and are found to satisfy the bound obtained by Beckner, Bialynicki-Birula, and Mycielski. These entropies for the first excited state, for different strengths of the potential well, are then numerically obtained. Interesting features of the entropy densities, owing their origin to the excited nature of the wave functions, are graphically demonstrated. We then compute the position space entropies of the coherent state of the Poeschl-Teller potential, which is known to show revival and fractional revival. Time evolution of the coherent state reveals many interesting patterns in the space-time flow of information entropy.

Quantum information entropies of the eigenstates and the coherent state of the Pöschl-Teller potential

A general algebraic procedure for constructing coherent states of a wide class of exactly solvable potentials, e.g., Morse and Poeschl-Teller, is given. The method, a priori, is potential independent and connects with earlier developed ones, including the oscillator-based approaches for coherent states and their generalizations. This approach can be straightforwardly extended to construct more general coherent states for the quantum-mechanical potential problems, such as the nonlinear coherent states for the oscillators. The time evolution properties of some of these coherent states show revival and fractional revival, as manifested in the autocorrelation functions, as well as, in the quantum carpet structures.

/pdf/coherent-states-for-exactly-solvable-potentials-3c8o1ifvdx.pdf

Coherent states for exactly solvable potentials

A new perspective on single and multi-variate differential equations

A novel method, connecting the space of solutions of a linear differential equation, of arbitrary order, to the space of monomials, is used for exploring the algebraic structure of the solution space. Apart from yielding new expressions for the solutions of the known differential equations, the procedure enables one to derive various properties of the orthogonal polynomials and functions, in a unified manner. The method of generalization of the present approach to the multi-variate case is pointed out and also its connection with the well-known factorization technique. It is shown that, the generating functions and Rodriguez formulae emerge naturally in this method.

Linear Differential Equations and Orthogonal Polynomials : a Novel Approach

We compute the one loop, O(θ) correction to the vertex in the noncommutative Chern-Simons theory with scalar fields in the fundamental representation. Emphasis is placed on the parity odd part of the vertex, since the same leads to the magnetic moment structure. We find that, apart from the commutative term, a θ-dependent magnetic moment type structure is induced. In addition to the usual commutative graph, cubic photon vertices also give a finite θ dependent contribution. Furthermore, the two two-photon vertex diagrams, that give zero in the commutative case yield finite θ dependent terms to the vertex function.

/pdf/induced-magnetic-moment-in-noncommutative-chern-simons-3pxc3jkal2.pdf

Induced magnetic moment in noncommutative Chern-Simons scalar QED

Text books on mathematical methods [1] and non-relativistic quantum mechanics [2] routinely take recourse to the method of series solution for solving linear differential equations encountered in various physical problems. The tediousness of the above procedure has often led to the search for alternate methods; the elegant raising and lowering operator approach for solving the harmonic oscillator problem serves as a classic example of these attempts. The other algebraic approach is group theoretical in nature and takes advantage of the symmetries of the problem at hand [2, 3]. The text book example of the Coulomb problem, where the angular momentum and the Runge-Lenz vectors combine to yield a complete algebraic description of the Hilbert space, demonstrates the efficacy of this procedure. Most of these methods fail to generalize to many-variable interacting systems, a field lately attracting considerable attention because of its relevance to many areas of physics [4, 5].

T. Shreecharan

Papers

Coherent states for exactly solvable potentials

A new perspective on single and multi-variate differential equations

Linear Differential Equations and Orthogonal Polynomials : a Novel Approach

Induced magnetic moment in noncommutative Chern-Simons scalar QED

A Novel Method to Solve Familiar Differential Equations and its Applications