Avinash Khare
Articles written in Pramana – Journal of Physics
Volume 6 Issue 5 May 1976 pp 312-322 Quantum Mechanics
Feynman diagram method for atomic collisions
Feynman diagram method of treating atomic collision problems in perturbation theory is presented and matrix elements are calculated for a number of processes. The result for the resonant charge transfer in hydrogen is identical to the well known OBK value. However, in processes like collisional ionisation, the results are different from those obtained by conventional methods.
Volume 14 Issue 5 May 1980 pp 327-341 Solid State Physics
Anharmonic oscillator model for first order structural phase transition
Exact solutions for the motion of a classical anharmonic oscillator in the potential
Volume 15 Issue 3 September 1980 pp 245-269 Statistical Mechanics
Classical φ^{6}-field theory in (1+1) dimensions. A model for structural phase transitions
The classical φ^{6}-field theory in (1+1) dimensions, is considered as a model for the first order structural phase transitions. The equation of motion is solved exactly; and the presence of domain wall (kink) solutions at and below the transition point, in addition to the usual phonon-like oscillatory solutions, is demonstrated. The domain wall solutions are shown to be stable, and their mass and energies are calculated. Above the transition point there exists exotic unstable kink-like solutions which takes the particle from one hill top to the other of the potential. The partition function of the system is calculated exactly using the functional integral method together with the transfer matrix techniques which necessitates the determination of the eigenvalues of a Schrödinger-like equation. Thus the exact free energy is evaluated which in the low temperature limit has a phonon part and a contribution coming from the domain wall excitations. It was shown that this domain wall free energy differs from that calculated by the use of the domain wall phenomenology proposed by Krumhansl and Schrieffer. The exact solutions of the Schrödinger-like equation are also used to evaluate the displacement-displacement, intensity-intensity correlation functions and the probability distribution function. These results are compared with those obtained from the phenomenology as well as the φ^{4}-field theory. A qualitative picture of the central peak observed in structural phase transitions is proposed.
Volume 15 Issue 5 November 1980 pp 501-505 Solid State Physics
The existence of a domain wall-like contribution to the free energy above the first order phase transition point is demonstrated for a system described by the ϕ^{6}-field theory in (1+1) dimensions.
Volume 48 Issue 2 February 1997 pp 537-553 Quantum Aspects Of Chaos
The phase of the Riemann zeta function
We, offer an alternative interpretation of the Riemann zeta function
Volume 49 Issue 1 July 1997 pp 41-64 Quantum Mechanics
Supersymmetry in quantum mechanics
In the past ten years, the ideas of supersymmetry have been profitably applied to many nonrelativistic quantum mechanical problems. In particular, there is now a much deeper understanding of why certain potentials are analytically solvable. In this lecture I review the theoretical formulation of supersymmetric quantum mechanics and discuss many of its applications. I show that the well-known exactly solvable potentials can be understood in terms of a few basic ideas which include supersymmetric partner potentials and shape invariance. The connection between inverse scattering, isospectral potentials and supersymmetric quantum mechanics is discussed and multi-soliton solutions of the KdV equation are constructed. Further, it is pointed out that the connection between the solutions of the Dirac equation and the Schrödinger equation is exactly same as between the solutions of the MKdV and the KdV equations.
Volume 49 Issue 2 August 1997 pp 199-204 Research Articles
Truncated harmonic oscillator and parasupersymmetric quantum mechanics
B Bagchi S N Biswas Avinash Khare P K Roy
We discuss in detail the parasupersymmetric quantum mechanics of arbitrary order where the parasupersymmetry is between the normal bosons and those corresponding to the truncated harmonic oscillator. We show that even though the parasusy algebra is different from that of the usual parasusy quantum mechanics, still the consequences of the two are identical. We further show that the parasupersymmetric quantum mechanics of arbitrary order
Volume 62 Issue 6 June 2004 pp 1201-1229
Local identities involving Jacobi elliptic functions
Avinash Khare Arul Lakshminarayan Uday Sukhatme
We derive a number of local identities involving Jacobi elliptic functions and use them to obtain several new results. First, we present an alternative, simpler derivation of the cyclic identities discovered by us recently, along with an extension to several new cyclic identities. Second, we obtain a generalization to cyclic identities in which successive terms have a multiplicative phase factor exp(2iπ/s), where
Volume 63 Issue 5 November 2004 pp 921-936
Connecting Jacobi elliptic functions with different modulus parameters
The simplest formulas connecting Jacobi elliptic functions with different modulus parameters were first obtained over two hundred years ago by John Landen. His approach was to change integration variables in elliptic integrals. We show that Landen’s formulas and their subsequent generalizations can also be obtained from a different approach, using which we also obtain several new Landen transformations. Our new method is based on recently obtained periodic solutions of physically interesting non-linear differential equations and remarkable new cyclic identities involving Jacobi elliptic functions.
Volume 73 Issue 2 August 2009 pp 375-385
Compactons in $\mathcal{PT}$-symmetric generalized Korteweg–de Vries equations
Carl M Bender Fred Cooper Avinash Khare Bogdan Mihaila Avadh Saxena
This paper considers the $\mathcal{PT}$-symmetric extensions of the equations examined by Cooper, Shepard and Sodano. From the scaling properties of the $\mathcal{PT}$-symmetric equations a general theorem relating the energy, momentum and velocity of any solitary-wave solution of the generalized KdV equation is derived. We also discuss the stability of the compacton solution as a function of the parameters affecting the nonlinearities.
Volume 73 Issue 2 August 2009 pp 387-395
New quasi-exactly solvable Hermitian as well as non-Hermitian $\mathcal{PT}$ -invariant potentials
Avinash Khare Bhabani Prasad Mandal
We start with quasi-exactly solvable (QES) Hermitian (and hence real) as well as complex $\mathcal{PT}$ -invariant, double sinh-Gordon potential and show that even after adding perturbation terms, the resulting potentials, in both cases, are still QES potentials. Further, by using anti-isospectral transformations, we obtain Hermitian as well as $\mathcal{PT}$ - invariant complex QES periodic potentials. We study in detail the various properties of the corresponding Bender–Dunne polynomials.
Volume 78 Issue 2 February 2012 pp 187-213 Research Articles
Solutions of several coupled discrete models in terms of Lamé polynomials of order one and two
Coupled discrete models abound in several areas of physics. Here we provide an extensive set of exact quasiperiodic solutions of a number of coupled discrete models in terms of Lamé polynomials of order one and two. Some of the models discussed are: (i) coupled Salerno model, (ii) coupled Ablowitz–Ladik model, (iii) coupled saturated discrete nonlinear Schrödinger equation, (iv) coupled $\phi^4$ model and (v) coupled $\phi^6$ model. Furthermore, we show that most of these coupled models in fact also possess an even broader class of exact solutions.
Volume 79 Issue 3 September 2012 pp 377-392
Solutions of several coupled discrete models in terms of Lamé polynomials of arbitrary order
Avinash Khare Avadh Saxena Apoorva Khare
Coupled discrete models are ubiquitous in a variety of physical contexts. We provide an extensive set of exact quasiperiodic solutions of a number of coupled discrete models in terms of Lamé polynomials of arbitrary order. The models discussed are: (i) coupled Salerno model, (ii) coupled Ablowitz–Ladik model, (iii) coupled $\phi^4$ model and (iv) coupled $\phi^6$ model. In all these cases we show that the coefﬁcients of the Lamé polynomials are such that the Lamé polynomials can be re-expressed in terms of Chebyshev polynomials of the relevant Jacobi elliptic function.
Volume 81 Issue 2 August 2013 pp 237-246 Research Articles
S Sree Ranjani A K Kapoor Avinash Khare P K Panigrahi
Quantum Hamilton–Jacobi formalism is used to give a proof for Gozzi’s criterion, which states that for eigenstates of the supersymmetric partners, corresponding to the same energy, the difference in the number of nodes is equal to one when supersymmetry (
Volume 85 Issue 5 November 2015 pp 915-928
We provide a systematic analysis of a prototypical nonlinear oscillator system respecting
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