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.

Exact solutions for the motion of a classical anharmonic oscillator in the potentialV(φ)=Bφ² − |A|φ⁴ +Cφ⁶ are obtained in (1 + 1) dimensions. Instanton-like solutions in (imaginary time) which takes the particle from one maximum of the potential to the other are obtained in addition to the usual oscillatory solutions. The energy dependence of the frequencies of oscillation is discussed in detail. This can be used as a model for the first order structural phase transition in the mean field approximation. The high and low temperature behaviour of the static susceptibility is obtained. Finally, a qualitative explanation is offered for the observed central peak in ferroelectrics like SrTiO₂.

The classical φ⁶-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 φ⁴-field theory. A qualitative picture of the central peak observed in structural phase transitions is proposed.

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 ϕ⁶-field theory in (1+1) dimensions.

The magnetic moments of uncharmed and charmed baryons are considered to arise through single-quark and two-quark transitions in a quark model. The magnetic moment operator is taken to transform as:T_β^α ˜aT₁¹, +bT₂² +cT₃³ +dT₄⁴, whereT_β^α are members of SU(4)20′-plet. The assumption, that the magnetic moment operator obtains contribution from the single and two-quark transitions, yields good results for the magnetic moment values of uncharmed baryons. Magnetic moments of charmed baryons can be expressed in terms of one parameter.

It is shown that the drift cyclotron loss cone instability can be suppressed by modulating electron density within the plasma. With the feedback in +90° phase the critical density gradient needed for the onset of the drift cyclotron loss cone instability increases approximately linearly with the gain. Typically with the gain of −50Ω_i the critical density gradient can be pushed up by as much as two orders of magnitude and minimum mirror plasma radius can be brought down in the same proportion.

Employing non-relativistic quasinuclear coloured quark model, which provides a unified description of mesons and baryons, masses of charmed andb-quark hadrons are studied. Various mass splittings are estimated, and mass relations among them are obtained.

The feedback suppression of flute instabilities by neutral beam injection is studied. It is shown that the available neutral beam current in mirror is sufficient to simulate the min-B field and thereby keep the flute modes in control.

The spin-correlation length is used to set up a RG analysis of the Hubbard model (within RPA). We demonstrate that an identical critical behaviour is obtained by performing the macroscopic renormalization group analysis with the antisymmetric Landau interaction parameter. The beta functions for the half-filled and quarter-filled band cases have been evaluated.

Static, non-magnetic impurities give rise to gap states in a doped Mott-Hubbard antiferromagnetic insulator. The spectral and spatial features of these gap states are discussed, and it is argued that these gap states are responsible for the observed local-moment behaviour in zinc-doped cuprates.

Experimental results on the measurement of current collected by an electrode immersed in a plasma for a pulsed negative bias are presented. The measured current is compared with a model based on the concept of an expanding capacitor. The scaling laws predicted by the model are verified for the measured current which agree each other. The paper emphasizes the role of displacement current in an expanding ion sheath.

We, offer an alternative interpretation of the Riemann zeta functionζ(s) as a scattering amplitude and its nontrivial zeros as the resonances in the scattering amplitude. We also look at several different facets of the phase of theζ function. For example, we show that the smooth part of theζ function along the line of the zeros is related to the quantum density of states of an inverted oscillator. On the other hand, for ℜs>1/2, we show that the memory of the zeros fades only gradually through a Lorentzian smoothing of the delta functions. The corresponding trace formula for ℜs≫1 is shown to be of the same form as generated by a one-dimensional harmonic oscillator in one direction along with an inverted oscillator in the transverse direction. Quite remarkably for this simple model, the Gutzwiller trace formula can be obtained analytically and is found to agree with the quantum result.

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.

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 orderp can also be rewritten in terms ofp supercharges (i.e. all of which obeyQ_i²=0). However, the Hamiltonian cannot be expressed in a simple form in terms of thep supercharges except in a special case. A model of conformal parasupersymmetry is also discussed and it is shown that in this case, thep supercharges, thep conformal supercharges along with HamiltonianH, conformal generatorK and dilatation generatorD form a closed algebra.

The three-band Hubbard model — both pure and with static non-magnetic impurities — has been studied within a self-consistent numerical Hartree-Fock (HF) scheme. The system shows nesting properties only in the absence of direct O-O hopping. Spin excitations in the system are gapless with the existence of a Goldstone mode in the broken-symmetry state. The variation of spinwave velocity with Cu-site Coulomb repulsion shows a (1/(2U_d)+(1/Δ)) dependence in the strong-coupling limit. Each non-magnetic impurity in the system gives rise to two gap states for a particular spin and the local moment produced is robust even at finite concentration of mobile hole doping. The gapless Goldstone mode is preserved even in case of unequal concentration of impurities on the two sublattices.

The vortex dynamics at microwave frequencies in YBa₂Cu₃O_7-δ (YBCO) films have been studied. We observe a peak in the microwave (4.88 and 9.55 GHz) surface resistance in some films in magnetic fields up to 0.8 T. This is associated with the ‘peak-effect’ phenomenon and reflects the order-disorder transformation of the flux line lattice near the transition temperature. Introduction of artificial pinning centers like columnar defects created as a result of irradiation with 200 MeV Ag ion (at a fluence of 4×10¹⁰ ions/cm²) leads to the suppression of the peak in films previously exhibiting ‘peak effect’.

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), wheres is any integer. Third, we systematize the local identities by deriving four local ‘master identities’ analogous to the master identities for the cyclic sums discussed by us previously. Fourth, we point out that many of the local identities can be thought of as exact discretizations of standard non-linear differential equations satisfied by the Jacobi elliptic functions. Finally, we obtain explicit answers for a number of definite integrals and simpler forms for several indefinite integrals involving Jacobi elliptic functions.

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.

In this paper we apply perturbative field-theoretic technique to helical turbulence. In the inertial range the kinetic helicity flux is found to be constant and forward. The universal constantK^H appearing in the spectrum of kinetic helicity was found to be 2.47.

The recent neutron scattering data for spin-wave dispersion in HoMnO₃ are well-described by an anisotropic Hubbard model on a triangular lattice with a planar (XY) spin anisotropy. Best fit indicates that magnetic excitations in HoMnO₃ correspond to the strong-coupling limit $U/t > \sim 15$, with planar exchange energy $J = 4t^{2}/U \simeq 2.5$ meV and planar anisotropy $\Delta U \simeq 0.35 $meV.

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.

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.

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.

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.

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 (SUSY) is unbroken and is zero when SUSY is broken. We also show that this proof is also applicable to the case, where isospectral deformation is involved.

We provide a systematic analysis of a prototypical nonlinear oscillator system respecting PT-symmetry, i.e., one of them has gain and the other an equal and opposite amount of loss. We first discuss various symmetries of the model. We show that both the linear system as well as a special case of the nonlinear system can be derived from a Hamiltonian, whose structure is similar to the Pais–Uhlenbeck Hamiltonian. Exact solutions are obtained in a few special cases. We show that the system is a superintegrable system within the rotating wave approximation (RWA). We also obtain several exact solutions of these RWA equations. Further, we point out a novel superposition in the context of periodic solutions in terms of Jacobi elliptic functions that we obtain in this problem. Finally, we briefly mention numerical results about the stability of some of the solutions.