A boundS_l is given for the number of bound statesn_i in thelth partial wave corresponding to a spherically symmetric potential in non-relativistic quantum mechanics. This bound is given by$$n_\iota = \mathop \smallint \limits_0^\infty |V_a (l,r)|^{1/2} dr/\pi + 1$$ whereV_a(l, r) is the attractive part of the effective potentialV(r)+l(l+1)/r². Extensive comparative study ofS_i and the Bargmann inequality is made.

A mathematical procedure to calculate the contribution to the reaction cross-section from a shell of radiusr and thickness Δ around the scattering centre within the frame work of a nuclear optical model is presented. The method is illustrated by describing graphically the regionwise absorption in nucleon-nucleus and nucleus-nucleus optical scattering. It is demonstrated that unlike in nucleon-nucleus scattering, in the nucleus-nucleus scattering volume absorptive optical potential, in general, does not imply that absorption is taking place in the entire nuclear volume; it is confined to mostly the surface region.

The similarities and differences between antiproton-nucleus scattering and heavyion-nucleus scattering are examined. It is found that the one-turning point approach viz Wentzel, Kramers and Brillouin (WKB) approximation can be applied to the analysis of antiproton-nucleus scattering. Using this approximation, a closed form expression for the nuclear phase shift is deduced from the corresponding expression for the heavy-ion scattering phase shift derived by Shastry and the method is illustrated by carrying out the cross-section calculation of$$\bar p + {}^{40}Ca$$ atE_lab = 46·8 MeV.

Using the relation between the number of bound states and the number of zeros of the radial eigen-functionψ(r), or equivalently, that ofφ(r)=rψ(r) in the range 0⩽r⩽∞, the upper bounds on the number of bound states generated by potentialV(r) in different angular momentum channels are obtained in three dimension. Using a similar procedure, the upper bound on the number of bound states in one dimension is also deduced. The analysis is restricted to a class of potentials for whichE=0 is the threshold. By taking a number of specific examples, it is demonstrated that both in one and three dimensions, the estimate of the upper bound obtained by this procedure is very close to or equal to the exact number of bound states. The correlation of the present method with the Levison’s theorem and WKB approximation is discussed.

Using the appropriate harmonic oscillator states and reasonable approximations, we construct coherent wavepackets corresponding to the solutions of the Klein-Gordon equation for the attractive potentialV(r)=−k/r, k>0, in two and three space dimensions. We deduce the corresponding classical limit in two dimension by requiring that the expectation value 〈r〉 of the radial variable is large. In the case of three dimensions, besides the condition of large 〈r〉, we make the uncertainty Δr=[〈r²〉 − 〈r〉²]^1/2 a minimum with respect to certain parameter of the wavepacket. We then investigate the trajectory traversed by the wavepacket in the classical limit. We find that the classical limit of this relativistic quantal problem gives, in the leading order, the same expression for the rate of motion of the perihelion as that given by the solution of the corresponding special relativistic classical dynamical problem. We also briefly discuss some of the subtle aspects of the classical limit of the relativistic quantal system, in general.

In this paper, we obtain reliable expressions to calculate the barrier and pocket positions of the real part of the effective phenomenological optical potential having Woods-Saxon form factor, for different partial waves. The comparison of the results obtained from these formulae, when compared with the numerical results obtained using Newton-Raphson iterative procedure are found to be quite accurate, with error less than 1%. We also obtain algebraic expressions for estimatingl_poc, the angular momentum at which the potential pocket vanishes, the accuracy of which is tested with the exact calculations, using self-consistent iterative procedures. These and other expressions deduced in this paper provide simple and useful methods for calculating critical parameters of heavy ion effective potentials like barrier and pocket positions, curvatures at the barrier and pocket positions,l_poc and the grazing angular momentuml_g to carry out the analysis of heavy ion scattering.

Using several illustrative examples, the nature of resonance poles and the corresponding zeroes of the s-waveS matrix is examined for several potentials having an absorptive pocket followed by a barrier. It is shown that even though the presence of absorption practically suppresses the manifestation of resonance in the elastic scattering cross section, the effect of the resonances generated by the absorptive pocket is more clearly manifested in the absorption cross section provided the barrier width is not too large. We further find that the signature of barrier top resonances are also more clearly manifested in the absorption cross section rather than in the elastic scattering cross section. These results have been interpreted in terms of complex resonance poles and corresponding zeroes of theS matrix. This implies that in complex potential scattering like heavy ion collisions, the reaction channel cross section peak is a more reliable signature of resonance phenomenon than the variation of the elastic channel cross section with energy.

We examine the behavior of transmission coefficient T across the rectangular barrier when attractive potential well is present on one or both sides and also the same is studied for a smoother barrier with smooth adjacent wells having Woods-Saxon shape. We find that presence of well with suitable width and depth can substantially alter T at energies below the barrier height leading to resonant-like structures. In a sense, this work is complementary to the resonant tunneling of particles across two rectangular barriers, which is being studied in detail in recent years with possible applications in mind. We interpret our results as due to resonant-like positive energy states generated by the adjacent wells. We describe in detail the possible potential application of these results in electronic devices using n-type oxygen-doped gallium arsenide and silicon dioxide. It is envisaged that these results will have applications in the design of tunneling devices.