Jost Function Research Articles

Trace formulas for Schrödinger operators with complex potentials on a half line

We consider Schr\"odinger operators with complex-valued decreasing potentials on the half-line. Such operator has essential spectrum on the half-line plus eigenvalues (counted with algebraic multiplicity) in the complex plane without the positive half-line. We determine trace formula: sum of Im part of these eigenvalues plus some singular measure plus some integral from the Jost function. Moreover, we estimate of sum of Im part of eigenvalues and singular measure in terms of the norm of potentials. In addition, we get bounds on the total number of eigenvalues, when the potential is compactly supported.

Off-shell Jost function for the Hulthén potential in all partial waves

A new expression for the Hulthén off-shell Jost function in all partial waves is constructed in its maximal reduced form. As a basic requirement the on-shell solutions are first developed by following the differential equation approach to the problem together with judicious exploitation of the properties of certain special functions of mathematical physics. Utilizing the properties of the Jost function, the binding energies and phase shifts for N-N and n-d systems are computed and found excellent agreement with standard data.

Jost function formalism based on the Hartree-Fock-Bogoliubov formalism

We formulate the Jost function formalism based on the Hartree-Fock-Bogoliubov (HFB) theory which has been used to represent the nature of the superfluidity of nucleus. The Jost function based on the HFB can give the analytic representation of the S-matrix for the nucleon elastic scattering targeting on the open-shell nucleus taking into account the pairing effect. By adopting the Woods-Saxon potential, we show the numerical results of S-matrix poles and their trajectories with varying the pairing strength in two cases of Fermi energies. The total cross sections in each cases for neutron elastic scattering are also analyzed, and we confirmed some staggering shapes or sharp resonances originated from the effect of pairing can be seen in the cross section.

On the Heisenberg condition in the presence of redundant poles of the S-matrix

For the same potential as originally studied by Ma (Phys. Rev., 71 (1947) 195) we obtain analytic expressions for the Jost functions and the residue of the S-matrix of both i) redundant poles and ii) the poles corresponding to true bound states. This enables us to demonstrate that the Heisenberg condition is valid in spite of the presence of redundant poles and a singular behaviour of the S-matrix for . In addition, we analytically determine the overall contribution of redundant poles to the asymptotic completeness relation, provided that the residue theorem can be applied. The origin of redundant poles and zeros is shown to be related to peculiarities of analytic continuation of a parameter of two linearly independent analytic functions.

Dynamics in the Szegő class and polynomial asymptotics

We introduce the Szegő class, Sz(E), for an arbitrary Parreau–Widom set E ⊂ ℝ and study the dynamics of its elements under the left shift. When the direct Cauchy theorem holds on ℂ\E, we show that to each J ∈ Sz(E) there is a unique element J′ in the isospectral torus, T E , so that the left-shifts of J are asymptotic to the orbit {J′ m } on T E . Moreover, we show that the ratio of the associated orthogonal polynomials has a limit, expressible in terms of Jost functions, as the degree n tends to ∞. This enables us to describe the large n behaviour of the orthogonal polynomials for every J in the Szegő class.

Difference equations with a point interaction

In this paper, we consider an impulsive second‐order difference equation on the whole axis. We determine eigenvalues, spectral singularities, continuous spectrum corresponding to this difference equation with an impulsive condition by using the asymptotic properties of Jost functions, and uniqueness theorems of analytic functions. Finally, we demonstrate that the impulsive difference equation has finite number of eigenvalues and spectral singularities with finite multiplicities under certain conditions.

Quantum decay law: critical times and the equivalence of approaches

Methods based on the use of Green’s functions or Jost functions and the Fock–Krylov method are apparently very different approaches to understand the time evolution of unstable states. We show that the two former methods are equivalent up to some constants and as an outcome find an analytic expression for the energy density of states in the Fock–Krylov amplitude in terms of the coefficients introduced in the Green’s functions and the Jost functions methods. This model-independent density is further used to obtain an analytical expression for the survival amplitude and study its behaviour at large times. Using these expressions, we investigate the origin of the oscillatory behaviour of the decay law in the region of the transition from the exponential to the non-exponential at large times. With the objective of understanding the failure of nuclear and particle physics experiments in observing the non-exponential decay law predicted by quantum mechanics for large times, we derive analytical formulae for the critical transition time, , from the exponential to the inverse power law behaviour at large times. Evaluating for some particle resonances and narrow nuclear states which have been tested experimentally to verify the exponential decay law, we conclude that the large time power law in particle and nuclear decay is hard to find experimentally.

Coarse graining pi pi scattering

We carry out an analysis of pi pi scattering in the IJ=00, 11 and 20 channels in configuration space up to a maximal center-of-mass energy sqrt{s}=1.4 GeV. We separate the interaction into two regions marked by an elementarity radius of the system; namely, a long distance region above which pions can be assumed to interact as elementary particles and a short distance region where many physical effects cannot be disentangled. The long distance interaction is described by chiral dynamics, where a two-pion-exchange potential is identified, computed and compared to lattice calculations. The short distance piece corresponds to a coarse grained description exemplified by a superposition of delta-shell potentials sampling the interaction with the minimal wavelength. We show how the so constructed non-perturbative scattering amplitude complies with the proper analytic structure, allowing for an explicit N/D type decomposition in terms of the corresponding Jost functions and fulfilling dispersion relations without subtractions. We also address renormalization issues in coordinate space and investigate the role of crossing when fitting the scattering amplitudes above and below threshold to Roy-equation results. At higher energies, we show how inelasticities can be described by one single complex and energy dependent parameter. A successful description of the data can be achieved with a minimal number of fitting parameters, suggesting that coarse graining is a viable approach to analyze hadronic processes.

Nonlocal Reductions of The Multicomponent Nonlinear Schrödinger Equation on Symmetric Spaces

The aim of this paper is to develop the inverse scattering transform (IST) for multi-component generalisations of nonlocal reductions of the nonlinear Schrodinger (NLS) equation with PT-symmetry related to symmetric spaces. This includes: the spectral properties of the associated Lax operator, Jost function, the scattering matrix and the minimal set of scattering data, the fundamental analytic solutions. As main examples, we use the Manakov vector Schr\"odinger equation (related to A.III-symmetric spaces) and the multi-component NLS (MNLS) equations of Kullish-Sklyanin type (related to BD.I-symmetric spaces). Furthermore, the 1- and 2-soliton solutions are obtained by using an appropriate modification of the Zakharov-Shabat dressing method. It is shown, that the MNLS equations of these types allow both regular and singular soliton configurations. Finally, we present here different examples of 1- and 2-soliton solutions for both types of models, subject to different reductions.

Bound states and spectral singularities of an impulsive Schrödinger equation

In this paper, we study the analytical properties of the Jost function of an impulsive Schrödinger equation. We also investigate the bound states and spectral singularities of this equation. We present some conditions on the potential function that guarantee that the impulsive Schrödinger equation has a finite number of bound states and spectral singularities with finite multiplicities.

Spectral properties of quadratic pencil of Schrödinger equations with transmission conditions

In this paper, we investigate analytic and asymptotic properties of the Jost functions of the quadratic pencil of Schrödinger equations with transmission conditions. We also study the eigenvalues and spectral singularities of these equations.

Functional determinants of radial operators in AdS2

We study the zeta-function regularization of functional determinants of Laplace and Dirac-type operators in two-dimensional Euclidean AdS2 space. More specifically, we consider the ratio of determinants between an operator in the presence of background fields with circular symmetry and the free operator in which the background fields are absent. By Fourier-transforming the angular dependence, one obtains an infinite number of one-dimensional radial operators, the determinants of which are easy to compute. The summation over modes is then treated with care so as to guarantee that the result coincides with the two-dimensional zeta-function formalism. The method relies on some well-known techniques to compute functional determinants using contour integrals and the construction of the Jost function from scattering theory. Our work generalizes some known results in flat space. The extension to conformal AdS2 geometries is also considered. We provide two examples, one bosonic and one fermionic, borrowed from the spectrum of fluctuations of the holographic frac{1}{4} -BPS latitude Wilson loop.

Singularities of the transmission coefficient and anomalous scattering by a dielectric slab

We prove the existence and describe the distribution on the complex plane of the singularities, resonant states (RSs), of the transmission coefficient in the problem of the plane wave scattering by a parallel-plate dielectric slab in free space. It is shown that the transmission coefficient has isolated poles all with nonzero imaginary parts that form countable sets in the complex plane of the refraction index or permittivity of the slab with the only accumulation point at infinity. The transmission coefficient never vanishes and anomalous scattering, when its modulus exceeds unity, occurs at arbitrarily small loss of the dielectric filling the layer. These results are extended to the cases of scattering by arbitrary multi-layer parallel-plane media. Connections are established between RSs, spectral singularities, eigenvalues of the associated Sturm–Liouville problems on the line, and zeros of the corresponding Jost function.

Sinc-interpolants in the energy plane for regular solution, Jost function, and its zeros of quantum scattering

In a remarkable note of Chadan [Il Nuovo Cimento 39, 697–703 (1965)], the author expanded both the regular wave function and the Jost function of the quantum scattering problem using an interpolation theorem of Valiron [Bull. Sci. Math. 49, 181–192 (1925)]. These expansions have a very slow rate of convergence, and applying them to compute the zeros of the Jost function, which lead to the important bound states, gives poor convergence rates. It is our objective in this paper to introduce several efficient interpolation techniques to compute the regular wave solution as well as the Jost function and its zeros approximately. This work continues and improves the results of Chadan and other related studies remarkably. Several worked examples are given with illustrations and comparisons with existing methods.

International Symposium on New Developments in Methods and Applications of Few-body Physics: In Memory of Professor SA Sofianos

PrefaceThe symposium was held in memory of Professor S A Sofianos, a distinguished physicist, who dedicated all his academic life to serving the University of South Africa and the Few-body physics community. Fourteen researchers from Brazil, Russia, Japan and South Africa presented their work at the symposium, while many others who were interested could not participate because of financial and visa challenges. Almost all the participants have collaborated with Professor S A Sofianos in their research.A wide range of few-body physics topics were discussed during the symposium. Nuclear interactions as well as bound and scattering processes formed the bases of all the discussions of the symposium. The long-standing Coulomb problem and the almost forgotten Marchenko inversion resurfaced. New developments in few-body methods such as the Artificial Neural Networks, Integrodifferential Equations Approach, Gaussian Expansion Method and the Faddeev Approach, which are applicable to stable nuclear systems, were discussed. Various three- and four-body nuclear systems were considered for illustrations of these methods. Developments in the Jost function and Jost matrix methods and their application to unstable exotic nuclear systems were also discussed. Professor S A Sofianos worked on all, and many more, of these methods and few-body systems.List of Committee, Sponsors and Participants are available in this pdf

Jost function method approach for study of unstable nuclei

We have developed the formalism of the Jost function method (JFM) to study unstable nuclei. We apply the JFM to the calculations, the partial decay widths in the coupled-channel problem, anti-bound (virtual) states, non-local potentials and Lagrange-mesh method. The results are compared with other methods, and we show the JFM formalism is useful to study the unstable nuclei.

Molecular hydrogen: a benchmark system for near threshold resonances in high partial waves

Benchmark reactions involving molecular hydrogen, such as H2 + D or H2 + Cl, provide the ideal platforms to investigate the effect of near threshold resonances (NTR) on scattering processes. Due to the small reduced mass of such systems, shape resonances in certain partial waves can provide features at scattering energies up to a few kelvin, achievable in current experiments. We explore the effects of NTRs on elastic and inelastic scattering for various partial waves ℓ (in the case of H2 + Cl for s-wave and H2 + D for p-wave scattering) and find that NTRs lead to a different energy scaling of the cross sections as compared to the well known Wigner threshold regime. We give a theoretical analysis based on Jost functions, and we explore the NTR effects for higher partial waves using a simple model that incorporates the key ingredients of coupled-channel scattering problems. The effect of long-range interactions is also discussed, with special attention paid to the appearance of an effective Wigner regime for elastic scattering in certain partial waves.

KPII: Cauchy–Jost function, Darboux transformations and totally nonnegative matrices**To the memory of our friend and colleague Peter P Kulish

Direct definition of the Cauchy–Jost (known also as Cauchy–Baker–Akhiezer) function is given in the case of a pure solitonic solution. Properties of this function are discussed in detail using the Kadomtsev–Petviashvili II equation as an example. This enables formulation of the Darboux transformations in terms of the Cauchy–Jost function and classification of these transformations. Action of Darboux transformations on Grassmanians—i.e. on the space of soliton parameters—is derived and the relation of the Darboux transformations with the property of total nonnegativity of elements of corresponding Grassmanians is discussed.

Particular spectral singularity in the continuum energies: a manifestation as resonances

We study the coalescence of two bound energy eigenstates embedded in the continuous spectrum of a real Hamiltonian HW and the singular point produced by this coalescence. At the singular point, the two unnormalized Jost eigenfunctions are no longer linearly independent but coalesce to give rise to a bound state eigenfunction embedded in the continuum. We disturb the potential VW by means of a truncation; this perturbation breaks the singular point in two resonances. The phase shift shows a jump of magnitude and the shape of the cross section shows two inverted peaks; this behaviour is due to the interference between the two nearly degenerate resonances and the background component of the Jost function.

Spectral properties of generalized eigenparameter dependent discrete Sturm-Liouville type equation

Let L denote the non-selfadjoint difference operator of second order with boundary condition generated in ℓ2 (ℕ) bywhere {an}n∈ℕ and {bn}n∈ℕ are complex sequences, γi, βi ∈ ℂ, i = 0, 1, 2, … , p and λ is a eigenparameter. Discussing the spectral properties of L, we investigate the Jost function, spectrum, the spectral singularities, and the properties of the principal vectors corresponding to the spectral singularities of L, if.