Jost Function Research Articles

Strong absorptions of thermal neutrons caused by virtual states

Abstract Cross sections of compound nucleus (CN) formation by S-wave neutron scattering, which began to be studied in a pioneering work by Feshbach et al., are reanalyzed by the optical model plus the method of Jost function (JFM), which is one of modern theories to handle bound and unbound continuum states in a unified manner. We have calculated the CN cross-section for a thermal neutron scattering (En = 25.3 meV) by about 300 stable targets. The enhancements in the CN cross section at the specific mass number region, A ∼ 10, 50 and 160, which are well-known results in the previous optical model analyses, are reproduced by the present calculation. The JFM calculation is applied to investigate the origin of the enhanced CN cross-section, and the poles in the scattering matrices (S-matrices) are systematically explored in the complex momentum plane. In JFM, the S-matrix poles representing the virtual states, which are different from the usual resonances described by the Breit-Wigner formula, appear around the origin (zero energy) in the momentum plane at the mass region with the enhanced CN cross-section. The present analysis of the S-matrix means that the virtual state strongly enhances the CN cross-section. We also discuss the present results in connection to giant resonances and compound nuclear resonances, which were regarded as the established theories to understand the enhanced absorption of the thermal neutron in the previous studies.

Application of the Jost Function for Calculation of the Ground State of the Lithium Atom

Application of the Jost Function for Calculation of the Ground State of the Lithium Atom

Inverse scattering properties with Levinson formula for cubic eigenparameter-dependent discrete Dirac problem

UDC 517.9 We present some properties of zeros of the Jost function and of the scattering function. Then the uniqueness of the kernel and the continuity of the scattering function with appropriate Levinson-type formula are investigated for the inverse discrete Dirac problem based on the cubic eigenparameter-dependent boundary condition.

Riemann–Hilbert problem for the defocusing Lakshmanan–Porsezian–Daniel equation with fully asymmetric nonzero boundary conditions

Abstract In this work, Riemann-Hilbert approach is demonstrated to investigate the defocusing Lakshmanan-Porsezian-Daniel equation with fully asymmetric non-zero boundary conditions. In contrast to the symmetry case, this paper focuses at the branch points related to the scattering problem rather than using the Riemann surfaces. For the direct problem, we analyse the Jost solution of lax pairs and some properties of scattering matrix, including two kinds of symmetries. The inverse problem at branch points can be presented, corresponding to the associated Riemann-Hilbert. Moreover, we investigate the time evolution problem and estimate the value of solving the solutions by Jost function. For the inverse problem, we construct it as a Riemann-Hilbert problem and formulate the reconstruction formula for the defocusing Lakshmanan-PorsezianDaniel equation. The solutions of the Riemann-Hilbert problem can be constructed by estimating the solutions. Finally, we work out the solutions with fully asymmetric non-zero boundary conditions precisely via utilizing Sokhotski-Plemelj formula and the square of the negative column transformation with the assistance of Riemann surfaces. These results are valuable for understanding physical phenomena and developing further applications of optical problems.

The Jost function and Siegert pseudostates from R-matrix calculations at complex wavenumbers

The Jost function and Siegert pseudostates from R-matrix calculations at complex wavenumbers

Multi-pole solitons and breathers for a nonlocal Lakshmanan-Porsezian-Daniel equation with non-zero boundary conditions

Utilizing the Riemann-Hilbert approach, we study the inverse scattering transformation, as well as multi-pole solitons and breathers, for a nonlocal Lakshmanan-Porsezian-Daniel equation with non-zero boundary conditions at infinity. Beginning with the Lax pair, we introduce the uniformization variable to simplify both the direct and inverse problems on the two-sheeted Riemann surface. In the direct scattering problem, we systematically demonstrate the analyticity, asymptotic behaviors and symmetries of the Jost functions and scattering matrix. By solving the corresponding matrix Riemann-Hilbert problem, we work out the multi-pole solutions expressed as determinants for the reflectionless potential. Based on the parameter modulation, the dynamical properties of the simple-, double- and triple-pole solutions are investigated. In the defocusing cases, we show abundant simple-pole solitons including dark solitons, anti-dark-dark solitons, double-hump solitons, as well as double- and triple-pole solitons. In addition, the asymptotic expressions for the double-pole soliton solutions are presented. In the focusing cases, we illustrate the propagations of simple-pole, double-pole, and triple-pole breathers. Furthermore, the multi-pole breather solutions can be reduced to the bright soliton solutions for the focusing nonlocal Lakshmanan-Porsezian-Daniel equation.

Multichannel scattering for the Schrödinger equation on a line with different thresholds at both infinities

The multichannel scattering problem for the stationary Schrödinger equation on a line with different thresholds at both infinities is investigated. The analytical structure of the Jost solutions and of the transition matrix relating the Jost solutions as functions of the spectral parameter is described. Unitarity of the scattering matrix is proved in the general case when some of the scattering channels can be closed and the thresholds can be different at left and right infinities on the line. The symmetry relations of the S -matrix are established. The condition determining the bound states is obtained. The asymptotics of the Jost functions and of the transition matrix are derived for a large spectral parameter.

Multi‐soliton solutions of the three‐level coupled Schrödinger Maxwell‐Bloch equations via the Riemann‐Hilbert approach

In this paper, a study of the coupled Schrödinger Maxwell‐Bloch equations, which described the propagation of two optical pulses in an optical medium with coherent three‐level atoms, is presented via the Riemann‐Hilbert approach. First, we performed a spectral analysis on the basis of Lax pair with the negative flow, and then the Jost function, scattering matrix, and their analytic and symmetric properties are given. Second, the Riemann‐Hilbert problem is established successfully through a standard dressing procedure via the Riemann‐Hilbert approach, and then the potential function related to the solution of the Riemann‐Hilbert problem is reconstructed. By introducing the special matrix functions, we can transform the irregular Riemann‐Hilbert problem into the regular one, which can be solved by the Plemelj formula. Finally, some applications are given to solve the Riemann‐Hilbert problem without reflection for the coupled Schrödinger Maxwell‐Bloch equation, and the multi‐soliton solutions are obtained explicitly. Moreover, the dynamic behaviors of some soliton solutions are discussed graphically by choosing appropriate parameters.

Contraction of variational principle and optical soliton solutions for two models of nonlinear Schrödinger equation with polynomial law nonlinearity

<abstract><p>Our study analyzes the two models of the nonlinear Schrödinger equation (NLSE) with polynomial law nonlinearity by powerful and comprehensible techniques, such as the variational principle method and the amplitude ansatz method. We will derive the functional integral and the Lagrangian of these equations, which illustrate the system's dynamic. The solutions of these models will be extracted by selecting the trial ansatz functions based on the Jost linear functions, which are continuous at all intervals. We start with the Jost function that has been approximated by a piecewise linear function with a single nontrivial variational parameter in three cases from a region of a rectangular box, then use this trial function to obtain the functional integral and the Lagrangian of the system without any loss. After that, we approximate this trial function by piecewise linear ansatz function in two cases of the two-box potential, then approximate it by quadratic polynomials with two free parameters rather than a piecewise linear ansatz function, and finally, will be approximated by the tanh function. Also, we utilize the amplitude ansatz method to extract the new solitary wave solutions of the proposed equations that contain bright soliton, dark soliton, bright-dark solitary wave solutions, rational dark-bright solutions, and periodic solitary wave solutions. Furthermore, conditions for the stability of the solutions will be submitted. These answers are crucial in applied science and engineering and will be introduced through various graphs such as 2D, 3D, and contour plots.</p></abstract>

A Study of The Scattering Properties of Eigenparameter- Dependent Matrix Difference Operator With Transmission Condition

In this paper, we set a transmission boundary value problem for a matrix valued difference equation on the semi axis. The main purpose of this study is to examine the properties of scattering solutions and scattering functions of this problem. Firstly, by giving the Jost solution and scattering solutions of this problem, we obtain the Jost function and the scattering function of the problem. We also investigate eigenvalues, spectral singularities, resolvent operator and continuous spectrum of this problem.

Jost function formalism with complex potential

The Jost function formalism is extended with use of the complex potential in this paper. We derive the Jost function by taking into account the dual state which is defined by the complex conjugate the complex Hamiltonian. By using the unitarity of the S-matrix which is defined by the Jost function, the optical theorem with the complex potential is also derived. The role of the imaginary part of the complex potential for both the bound states and the scattering states is figured out. The numerical calculation is performed by using the complex Woods-Saxon potential, and some numerical results are demonstrated to confirmed the properties of extended Jost function formalism.

Low Energy Analysis of Phase Shifts and Cross-Section for n – 3He System

"Considering the Hulth´en potential as the nuclear part of interaction regular solution is derived by exploiting the Frobenius method. The Jost function is constructed by utilizing its integral representation in terms of regular solution. The scattering phase shifts and cross-sections are calculated for the n−3He system from the on-shell Jost function. The results obtained are in close conformity with the previous works that exist in the literature."

Jost function analysis of elastic and inelastic nucleon–nucleon scattering with a new model potential

The analysis of nucleon–nucleon elastic and inelastic scattering data is presented by considering the combined interaction of nuclear and electromagnetic potentials of equal range. The exact analytical off-shell solutions of the nonrelativistic Schrödinger equation for the effective potential with the centrifugal term are investigated using ordinary differential equation approach. Numerical results of scattering phase shifts, differential cross-sections and Transition matrices are in sensible conformity with previous works.

Double and triple-pole solutions for the third-order flow equation of the Kaup-Newell system with zero/nonzero boundary conditions

In this work, the double and triple-pole solutions for the third-order flow equation of Kaup-Newell system (TOFKN) with zero boundary conditions (ZBCs) and non-zero boundary conditions (NZBCs) are investigated by means of the Riemann-Hilbert (RH) approach stemming from the inverse scattering transformation. Starting from spectral problem of the TOFKN, the analyticity, symmetries, asymptotic behavior of the Jost function and scattering matrix, the matrix RH problem with ZBCs and NZBCs are constructed. Then the obtained RH problem with ZBCs and NZBCs can be solved in the case of scattering coefficients with double or triple zeros, and the reconstruction formula of potential, trace formula as well as theta condition are also derived correspondingly. Specifically, the general formulas of N-double and N-triple poles solutions with ZBCs and NZBCs are derived systematically by means of determinants. The vivid plots and dynamics analyses for double and triple-pole soliton solutions with the ZBCs as well as double and triple-pole interaction solutions with the NZBCs are exhibited in details. Compared with the most classical second-order flow Kaup-Newell system, we find the third-order dispersion and quintic nonlinear term of the Kaup-Newell system change the trajectory and velocity of solutions. Furthermore, the asymptotic states of the 1-double poles soliton solution and the 1-triple poles soliton solution are analyzed when t tends to infinity.

Analytical solution for nuclear plus atomic Hulthén potential — A perturbative approach

In this work, we construct the analytical solution of the Schrödinger equation for a combined nuclear plus atomic Hulthén potential with different range parameters, using the Frobenius method. The atomic Hulthén potential acts as the screened Coulomb potential to represent the very short-range electromagnetic interaction. It is intended to emphasize how the impact of the combined potential in these situations is routinely examined within the context of nuclear physics. The Jost function is calculated from its integral representation in terms of the regular solution. From the phase of the Jost function, the scattering phase shifts for different partial waves, and further differential scattering cross-section and total cross-sections are calculated for alpha-proton and proton–proton systems. On comparing with the existing experimental data, we conclude that the results obtained are in close conformity with the previous works that exist in the literature.

An inverse scattering problem for eigenparameter-dependent discrete Dirac system with Levinson formula

In this paper, we have considered an inverse scattering problem for discrete Dirac operator L with eigenparameter dependent boundary condition. At the outset, the Jost solution and the Jost function of L are given. Then the zeros of the Jost function and some properties of the scattering function are examined. After finding the scattering dataset and the main equation for the operator L, the uniqueness of the kernel, continuity of the scattering function and the appropriate Levinson type formula are obtained.

Corrigendum to “Inverse scattering problem with Levinson formula for eigenparameter‐dependent discrete Sturm‐Liouville equation”

There are three oversights in the zeros of the Jost function in Lemma 3.3 (page 1469), in the Delta function in Lemma 3.4 (page 1471), and in the continuity of the scattering function at zero in Theorem 4.2 (page 1473). Therefore, we have corrected Lemma 3.3, Lemma 3.4, and Theorem 3.5 with their proofs and Section 4.

Inverse scattering transform for the two‐component Gerdjikov–Ivanov equation with nonzero boundary conditions: Dark‐dark solitons

AbstractThe two‐component Gerdjikov–Ivanov equation with nonzero boundary conditions is studied by the inverse scattering transform. A fundamental set of analytic eigenfunctions is obtained with the aid of the associated adjoint problem. Three symmetry conditions are discussed to curb the scattering data. The behavior of the Jost functions and the scattering matrix at the branch points is discussed. The inverse scattering problem is formulated by a matrix Riemann–Hilbert problem. The trace formula in terms of the scattering data and the so‐called asymptotic phase difference for the potential are obtained. The solitons classification is described in detail. When the discrete eigenvalues lie on the circle, the dark‐dark soliton is obtained for the first time in this work. And the discrete eigenvalues off the circle generate the dark‐bright, bright‐bright, breather‐breather, M(‐type)‐W(‐type) solitons, and their interactions.

Breather solutions to the AB system with non-zero background

On the basis of the non-zero boundary condition to the AB system, we derive the asymptotic Lax pair and study its spectral analysis, from which we obtain the properties of the modified Jost functions including the analyticity, asymptotic behavior and symmetric relation, and construct the corresponding Riemann-Hilbert problem. The potentials can be reconstructed according to the asymptotic expansion of the sectional analytic function and the Plemelj formulae. As an application, we study the reflectionless case of the AB system and obtain its breather solutions with some figures showing the dynamic behaviors.

Scattering properties of the nonlinear eigenvalue‐dependent Sturm‐Liouville equations with sign‐alternating weight and jump condition

This paper aims to investigate the scattering function and discrete spectrum of the impulsive Sturm‐Liouville type differential operator with a turning point and nonlinear eigenparameter‐dependent boundary condition. Using hyperbolic type representations of the fundamental solutions, the operator's discrete spectrum was constructed. We presented asymptotic equations for the fundamental solutions and the Jost function. Finally, we stated an example to demonstrate the paper's major points.