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A new hybrid eighth-algebraic-order two-step method with phase-lag of order ten is developed for computing elastic scattering phase shifts of the one-dimensional Schrödinger equation. Based on this new method and on the method developed recently by Simos we obtain a new variable-step procedure for the numerical integration of the Schrödinger equation. Numerical results obtained for the integration of the phase shift problem for the well known case of the Lenard–Jones potential show that this new method is better than other finite difference methods.

A family of new hybrid four-step tenth algebraic order methods with phase-lag of order fourteen is developed for accurate computations of the radial Schrödinger equation. Numerical results obtained for the integration of the phase shift problem for the well known case of the Lennard-Jones potential and for the numerical solution of the coupled equations arising from the Schrödinger equation show that these new methods are better than other finite difference methods.

We discuss a manyparticle quantum system, which is obtained by adding some nonhermitian but PT (i.e. combined parity and time reversal) invariant interaction to the A_N-1 rational Calogero model without confining potential. This model gives rise to scattering states with continuous real spectrum. The scattering phase shift is determined through the exchange statistics parameter. We find that, unlike the case of the usual Calogero model, the exclusion and exchange statistics parameter differ from each other in the presence of PT invariant interaction.

Phase shifts produced by two supersymmetric partner potentials satisfy a simple relation. One expects that the phase shift can be obtained algebraically, if the potential is shape invariant. We show that this is true only for a special class of shape invariant potentials. Among known shape invariant potentials, only the Coulomb potential satisfy this requirement. The Coulomb phase shift can be obtained analytically, which agrees with the well-known standard result.

We consider the quantum scattering problem of a relativistic particle in (2 + 1)-dimensional cosmic string spacetime under the influence of a nontrivial boundary condition imposed on the solution of the Klein–Gordon equation. The solution is then shifted as consequence of the nontrivial boundary condition and the role of the phase shift is to produce an Aharonov–Bohm-like effect. We examine the connection between this phase shift and the electromagnetic and gravitational analogous of the Aharonov–Bohm effect and compare the present results with previous ones obtained in the literature, also considering non-relativistic cases.

We use the Tridiagonal Representation Approach (TRA) to obtain exact scattering and bound states solutions of the Schrödinger equation for short-range inverse-square singular hyperbolic potentials. The solutions are series of square integrable functions written in terms of the Jacobi polynomial with the Wilson polynomial as expansion coefficients. The series is finite for the discrete bound states and infinite but bounded for the continuum scattering states.

In this paper, both the fifth-order Runge–Kutta numerical scheme and binary collision approximation are used to study the phase shift. Both numerical and theoretical results are shown that the solitary wave after head-on collision propagates along the chain behind the reference wave in both even and odd numbers of grain chains. It is the well-known feature of the appearance of the phase shift. Those results are in agreement with the experimental results. Furthermore, it is found that the phase shift is not only related to the collision position of the waves, but also to the position where the time is measured. The value of phase shift increases nonmonotonously with increasing the velocity of the opposite propagation of the wave. Binary collision approximation is applied to analyze the phase shift, and it is found that theoretical results agree well with numerical results, especially in the case of phase shift in odd chain.

In this paper, we investigate head-on and overtaking collisions of two solitary waves propagating in two-dimensional (2D) hexagonal packing confined in a rectangular region, using the fifth-order Runge–Kutta numerical scheme. In the head-on collision, numerical results indicate that the velocity amplitude of the solitary waves and the second solitary waves is proportional to the impacting velocity of the solitary waves. The relationship between phase shift and impacting velocity satisfies the equation $\mathrm{\Delta }t\sim {(V_{\mathrm{\text{Le}}})}^P$. For head-on collisions of the waves with an equal amplitude of velocity, this power exponent is $P=-0.202$, which can be well predicted by the binary collision approximation theory. For the case of head-on collisions between the waves with unequal amplitudes of solitary waves, this power exponent becomes $P=-0.656$. During an overtaking collision, the fast solitary wave has a positive phase shift, and the slow solitary wave has a negative phase shift. The phase shift of the fast solitary wave also has a power exponential relationship with the impacting velocity. The two waves either form a single peak in transition, or remain separated throughout the encounter with a double-peak formation depending on the ratio of the impacting velocity of solitary waves. The critical condition between the two regimes is consistent with those obtained by the Korteweg–de Vries model.

The Josephson equation with constant current and sinusoidal forcings and a phase shift is investigated in detail: the existence and the bifurcations of harmonics and subharmonics under small perturbations are given, by using the second-order averaging method and Melnikov function; the influence on bifurcations of periodic or subharmonics as the phase shift varies is considered; some numerical simulation results are reported in order to prove our theoretical predictions.

The buck converter has been known to exhibit chaotic behavior in a wide parameter range. In this paper, the resonant parametric perturbation method is applied to control chaos in a voltage-mode controlled buck converter. In particular, the effects of phase shift and frequency mismatch in the perturbing signal are studied. It is shown that the control power can be significantly reduced if the perturbation is applied with an appropriate phase shift. Moreover, when frequency mismatch is inevitable, intermittent chaos occurs, but effective control can still be accomplished at the expense of raising the control power. Analysis, simulations and experimental measurements are presented to provide theoretical and practical evidences for the proposed control method.

Pendulum equation with a phase shift, parametric and external excitations is investigated in detail. By applying Melnikov's method, we prove the criteria of existence of chaos under periodic perturbation. Numerical simulations, including bifurcation diagrams of fixed points, bifurcation diagrams of the system in three- and two-dimensional spaces, homoclinic and heteroclinic bifurcation surfaces, Maximum Lyapunov exponents (ML), Fractal Dimension (FD), phase portraits, Poincaré maps are plotted to illustrate the theoretical analysis, and to expose the complex dynamical behaviors including the onset of chaos, sudden conversion of chaos to period orbits, interior crisis, periodic orbits, the symmetry-breaking of periodic orbits, jumping behaviors of periodic orbits, new chaotic attractors including two-three-four-five-six-eight-band chaotic attractors, nonchaotic attractors, period-doubling bifurcations from period-1, 2, 3 and 5 to chaos, reverse period-doubling bifurcations from period-3 and 5 to chaos, and so on.

By applying the second-order averaging method and Melnikov's method, we obtain the criteria of existence of chaos in an averaged system under quasi-periodic perturbation for Ω = nω + ϵν, n = 1, 2, 4, but cannot prove the criteria of existence of chaos in the averaged system under quasi-periodic perturbation for Ω = nω + ϵν, n = 3, 5 – 15, by Melnikov's method, where ν is not rational to ω. By using numerical simulation, we have verified our theoretical analysis and studied the effect of parameters of the original system on the dynamical behaviors generated under quasi-periodic perturbations, such as the onset of chaos, jumping behaviors of quasi-periodic orbits, interleaving occurrence of chaotic behaviors and nonchaotic behaviors, interior crisis, quasi-periodic orbits to chaotic attractors, sudden conversion of chaos to quasi-periodic behaviors, nonchaotic attractors, and so on. However, we did not find period-doubling and reverse period-doubling bifurcations. We found that the dynamical behaviors under quasi-periodic perturbations are different from that under periodic perturbations, and the dynamics with a phase shift are different from the dynamics without phase shift.

The nucleon–nucleon (NN) potential is the residual interaction of the strong interaction in the low-energy region and is also the fundamental input to the study of atomic nuclei. Based on the nonperturbative properties of the quantum chromodynamics (QCD), NN potential is not yet directly accessible from QCD theory. Therefore, various models of NN interactions have been constructed based on Yukawa’s meson exchange pictures since the 1930s, including one-boson-exchange models, coordinate operator models and chiral effective field models. Analysis of extensive NN scattering data has shown that the two-body nuclear force exhibits a short-range repulsion and intermediate-range attraction, and decays rapidly with increasing distance. A series of charge-dependent high-precision NN interactions have been further developed in the past 30 years such as the AV18 potential, CD-Bonn potential, pvCD-Bonn potentials and the chiral effective nuclear potentials with momentum expansion up to the fifth order. In this work, the phase shifts at different channels, the cross-sections, the entanglement entropy in spin space, and the equations of state of symmetric nuclear matter and pure neutron matter from these high-precision NN interactions are calculated and systematically compared. It can be found that they have significant differences in the cases with high-angular momentum, high-laboratory energy and high-density regions.

In the technology of optical surface measurement, the phase-shifting shear interferometer had been widely used. In measurement, the phase-shifting was introduced in time domain, and the measurement system adopted double optical path for change phase-shifting and shear, so the measurement accuracy was limited. To decrease error, a commonpath phase-shifting shear system based on birefringent optical devices was introduced. In this system, the lateral shear and phase-shifting 0, π/2, π, 3π/2 was introduced in the commonpath optical system. Because the birefringent optical devices can introduce shear in the collinear optical system, a new Jones matrix was established. Based on the Jones matrix theory, the principle theory was analyzed in detail. In the experiment, an optical surface was measured with this system. The theoretical analysis and experiment results demonstrate the feasibility of this approach.

The present research is devoted to the investigation of the spin-dependant Goos–Hanchen phase shift in quantum nanodevice. This nanodevice is modeled as semiconducting quantum dot coupled to two ferromagnetic leads. The spin transport through such nanodevice is conducted under the effect of both magnetic field and the photon energy of the induced ac-field. The angle of incidence of electrons is taken into account. Results show that the Goos–Hanchen phase shift of spin-up electrons is different from that of spin-down electron. Also, spin polarization and giant magneto-resistance are strongly depending on the angle of incidence of electrons and the photon energy of the induced ac-field. The present model could realize experimentally the spin beam splitter and spin filter needed for spin qubits and quantum information processing.