Attractive Delta-function Potential Research Articles

Quantum quench dynamics of the attractive one-dimensional Bose gas via the coordinate Bethe ansatz

We use the coordinate Bethe ansatz to study the Lieb–Liniger model of a one-dimensional gas of bosons on a finite-sized ring interacting via an attractive delta-function potential. We calculate zero-temperature correlation functions for seven particles in the vicinity of the crossover to a localized solitonic state and study the dynamics of a system of four particles quenched to attractive interactions from the ideal-gas ground state. We determine the time evolution of correlation functions, as well as their temporal averages, and discuss the role of bound states in shaping the postquench correlations and relaxation dynamics.

Electron spectrum of a single-wall carbon nanotube in the framework of the nonlinear Schrödinger equation

The electron spectrum of a single-wall carbon metal nanotube is analyzed numerically. The interaction of a free electron with atomic ions and bound electrons is approximated by an attractive delta-function potential in the single-particle Schrodinger equation. The interaction of an electron with other free electrons is presented by the Hartree nonlinear repulsive short-range potential.

Infinitely many singular interactions on noncompact manifolds

We show that the ground state energy is bounded from below when there are infinitely many attractive delta function potentials placed in arbitrary locations, while all being separated at least by a minimum distance, on two dimensional non-compact manifold. To facilitate the reading of the paper, we first present the arguments in the setting of Cartan–Hadamard manifolds and then subsequently discuss the general case. For this purpose, we employ the heat kernel techniques as well as some comparison theorems of Riemannian geometry, thus generalizing the arguments in the flat case following the approach presented in Albeverio et al. (2004).

Conditions governing the generalisation of threshold bound states by N attractive delta potentials in one and three dimensions

Abstract This paper proves that for N attractive delta function potentials the number of bound states (Nb) satisfies 1 ≤ N b ≤ N in one dimension (1D), and is 0 ≤ N b ≤ N in three dimensions (3D). Algebraic equations are obtained to evaluate the bound states generated by N attractive delta potentials. In particular, in the case of N attractive delta function potentials having same separation a between adjacent wells and having the same strength λV, the parameter g=λVa governs the number of bound states. For a given N in the range 1–7, both in 1D and 3D cases the numerical values of gn, where n=1,2,..N are obtained. When g=gn, Nb ≤ n where Nb includes one threshold energy bound state. Furthermore, gn are the roots of the Nth order polynomial equations with integer coefficients. Based on our numerical calculations up to N=40, even when N becomes large, 0 ≤ g n ≤ 4 and $\frac{{\Sigma g_n }} {N} \simeq 2 $ and this result is expected to be generally valid. Thus, for g > 4 there will be no threshold or zero energy bound state, and if g≈ 2 for a given large N, the number of bound states will be approximately N/2. The empirical formula gn = 4/[1+exp((N 0 − n)/β)] gives a good description of the variation of gn as a function of n. This formula is useful in estimating the number of bound states for any N and g both in 1D and 3D cases.

The attractive nonlinear delta-function potential

We solve the one-dimensional nonlinear Schrödinger equation for an attractive delta-function potential at the origin, [(p2/2m)−Ωδ(x)|φ(x)|α]φ(x)=Eφ(x), and obtain the bound state in closed form as a function of the nonlinear exponent α. The bound state probability profile decays exponentially away from the origin, with a profile width that increases monotonically with α, becoming an almost completely extended state when α→2−. At α=2, the bound state suffers a discontinuous change to a delta function-like profile. A further increase of α increases the width of the probability profile, although the bound state is stable only for α<2. The transmission of plane waves across the potential increases monotonically with α and is insensitive to the sign of the opacity Ω.

Effective adiabatic approximation in the problem of three bodies coupled via short-range potentials

The effective adiabatic approximation is constructed for the problem of three bodies on a straight line that are coupled via short-range attractive delta-function potentials. It is shown that, in this system, there arise a nonlocal momentum-dependent long-range effective potential and a polarization potential. A lower bound on the binding energy of the system is obtained to a relative precision of about 10−6. It is shown that, to within 0.03%, this approximation yields a correct asymptotic behavior of solutions and a correct behavior of the phase shift for elastic scattering at relative momenta below the three-body-breakup threshold. A local convergence of the adiabatic expansion in a finite interval of the radial variable is demonstrated.

Collisions between a small number of photons in a nonlinear Kerr medium

The quantum field theory for the one-dimensional Schrödinger equation with attractive delta function potentials is used for treating the dynamics of collisions between a small number of photons in a nonlinear Kerr medium. The present theoretical analysis takes into account both bound states and dispersive continuum states and the general theory is applied, especially for the systems of diphotons and three-photons.

Dimensional expansion for the delta-function potential

This paper examines an attractive delta-function potential in D-dimensional space, where D is regarded as an arbitrary complex number. The renormalized scattering amplitude is calculated for arbitrary D and is shown to be finite as . Dimensional expansions about D = 0 and D = 2 are obtained; these expansions have a non-zero radius of convergence.

Supersymmetric one-parameter strict isospectrality for the attractive potentials

The Schr\"odinger equation with attractive delta potential has been previously studied in the supersymmetric quantum mechanical approach by a number of authors, but they all used only the particular superpotential solution. Here, we introduce a one-parameter family of strictly isospectral attractive delta function potentials, which is based on the general superpotential (general Riccati) solution, we study the problem in some detail and suggest possible applications

Calculating the rest tension for a polymer of string bits.

We explore the application of approximation schemes from many body physics, including the Hartree-Fock method and random phase approximation (RPA), to the problem of analyzing the low energy excitations of a polymer chain made up of bosonic string bits. We accordingly obtain an expression for the rest tension $T_0$ of the bosonic relativistic string in terms of the parameters characterizing the microscopic string bit dynamics. We first derive an exact connection between the string tension and a certain correlation function of the many-body string bit system. This connection is made for an arbitrary interaction potential between string bits and relies on an exact dipole sum rule. We then review an earlier calculation by Goldstone of the low energy excitations of a polymer chain using RPA. We assess the accuracy of the RPA by calculating the first order corrections. For this purpose we specialize to the unique scale invariant potential, namely an attractive delta function potential in two (transverse) dimensions. We find that the corrections are large, and discuss a method for summing the large terms. The corrections to this improved RPA are roughly 15\%.

Perfect localized basis functions for solids: chemical pseudopotentials and the Kronig-Penney model

Anderson's chemical pseudopotential scheme is supposed to define a set of highly localized basis functions in terms of which the eigenstates of a solid can be expanded exactly. This sounds good in principle but little is known about the method in practice and it has not even been established that the basis functions always exist. This paper discusses some of the general properties of localized basis sets spanning a band of Bloch eigenstates and then looks at the Kronig-Penney model as an example. It is shown that the chemical pseudopotential equation has no solutions when the strength P of the attractive delta function potentials is less than about 1.4018, and that the basis functions generated are never unique when they do exist. Fortunately, it turns out that there is a generalized version of the chemical pseudopotential equation that can be solved no matter how weak the potentials, although the non-uniqueness of the basis functions remains. The solutions resemble atomic orbitals in the extreme tight-binding limit (although even then there are alternatives) but this is not true in general and so chemical pseudopotential theory should not be taken as a justification for using atomic orbitals as basis functions.

Two singular potentials: The space-splitting effect

The quantum mechanical problem of a particle moving in one dimension (1D) under the influence of a singular potential is reexamined. An attractive delta function potential of infinite strength and the 1D Coulomb potential is considered in particular. In both cases the singularity acts as an impenetrable barrier, thus dividing space into two independent regions. Analysis of the scattering states gives results in agreement with this conclusion. The implications that the bound states of the 1D Coulomb potential cannot be classified according to parity, even though the potential is symmetric, is also discussed herein.

Electron photodetachment from a Dirac bubble potential. A model for the fullerene negative ion CC 60−

A model for the fullerene anion C 60 − is proposed in which the outer electron moves in an attractive deltafunction potential in the shape of a spherical bubble. Exact solutions for this “Dirac bubble potential” were previously worked out. On the basis of this model, the photodetachment cross-section and its angular distribution are computed as functions of photoelectron energy, for the electron in both S-like and P-like bound states. Secondary maxima and minima in the cross section appear at higher energy values, somewhat analogous to the scattering of radiation from a conducting sphere.

Solution of the Chandler-Gibson equations for a three-body test problem.

The Chandler-Gibson (CG) {ital N}-body equations are tested by considering the problem of three nonrelativistic particles moving on a line and interacting through attractive delta-function potentials. In particular, the input Born and overlap matrix-valued functions are evaluated analytically, and the CG equations are solved using a {ital B}-spline collocation method. The computed scattering matrix elements are within 0.5% of the known exact solutions, and the corresponding scattering probabilities are within 0.001% of the exact probabilities, both below and above the 3-body breakup threshold. These results establish that the CG method is practical, as well as theoretically correct, and may be a valuable approach for solving certain more complicated {ital N}-body scattering problems.

Scattering from an attractive delta-function potential

A one−dimensional calculation is carried out to determine the phase shifts for scattering from an attractive delta−function potential. (AIP)

Comparison of multiple scattering approximations in a three body problem

An analysis of the nonrelativistic quantum mechanical problem of three particles of equal mass interacting via attractive delta function potentials in one dimension is carried out. Following the methods of McGuire and Dodd with two body potentials of equal strength, exact matrix elements for the process of elastic and rearrangement scattering are rederived and some mistakes occurring in their papers have been corrected. Then expressions are obtained for transition matrix elements for these processes in the first order using Newton, Born, Newton–Faddeev, and Lovelace–Faddeev formulation. Numerical results obtained are compared with the exact results. It is found that above breakup threshold the Newton–Faddeev scheme gives better results in comparison to other schemes for elastic as well as rearrangement scattering. Thus, this paper provides us with a comparative analysis of several multiple scattering expansions available in the literature.

A new expansion method in the Feynman path integral formalism: Application to a one-dimensional delta-function potential

An expansion method in the path-integral formulation of quantum mechanics, as proposed in a previous paper, is extended to account for states with odd parity. The method is tested on the case of a one-dimensional attractive delta-function potential and the well-known solutions for both bound and scattering states are obtained analytically by an exact summation of the expansion series. The applicability of the method is discussed and it is shown how the energy spectrum could be determined by means of Stieltjes theory of moments.

Interacting Fermions in One Dimension. II. Attractive Potential

The exact energies and wavefunctions for a system of (N − 1) one-dimensional fermions all of the same spin and one fermion of the opposite spin are calculated in the large volume, finite density limit, when the particles interact via an attractive delta function potential. It is found that the attractive potential gives rise to a bound state, but, in spite of the presence of this bound state, all of the physical properties which are calculated (ground-state energy, effective mass of a certain class of excitations, etc.) are analytic continuations in the coupling constant of the corresponding results in the repulsive case. In addition, it is possible to have eigenstates which do not have the bound state present. These excited states are also discussed and are found to exhibit a negative effective mass and to modify the particle density at very large distances from the different particle.

Application of the Rayleigh-Schrödinger Perturbation Theory to the Delta Function Potential

E. P. Wigner has shown that although there are reasons why one might hope to succeed, one cannot in fact derive the binding energy of the hydrogen atom by second-order perturbation theory applied to a free particle in a box. To further clarify the reasons for this failure we here consider the one-dimensional attractive delta function potential.