Zero-range Potential Research Articles

Integrable zero-range potentials in a plane

We examine general statements in the Wronskian representation of Darboux transformations for plane zero-range potentials. Such expressions naturally contain scattering problem solution. We also apply Abel theorem to Wronskians for differential equations and link it to chain equations for Darboux transforms to fix conditions for further development of the underlying distribution concept. Moutard transformations give a convenient insight into the problem that allows one to formulate general assertions and give complete description of a point potential in a limit of long waves.

Single-Particle Momentum Distributions for Bosonic Trimer States in Two and Three Dimensions

For a system formed by three bosons interacting by a pairwise zero-range potential we show how the functional form of the asymptotic momentum distribution changes drastically when passing from bi (2D) to tridimensional (3D) regime, mainly affected by the absence/presence of the Efimov effect in 2D/3D. The spectator functions and the momentum distribution are calculated analytically for both regimes.

Contact parameters in two dimensions for general three-body systems

We study the two-dimensional three-body problem in the general case of three distinguishable particles interacting through zero-range potentials. The Faddeev decomposition is used to write the momentum-space wave function. We show that the large-momentum asymptotic spectator function has the same functional form as derived previously for three identical particles. We derive the analytic relations between the three different Faddeev components for the three distinguishable particles. We investigate the one-body momentum distributions both analytically and numerically and analyze the tail of the distributions to obtain two- and three-body contact parameters. We specialize from the general cases to the examples of two identical, interacting or non-interacting, particles. We find that the two-body contact parameter is not a universal constant in the general case and show that the universality is recovered when a subsystem is composed of two identical non-interacting particles. We also show that the three-body contact parameter is negligible in the case of one non-interacting subsystem compared to the situation where all the subsystems are bound. As an example, we present the results for mixtures of lithium with two cesium or two potassium atoms, which are systems of current experimental interest.

Multicomponent Strongly Interacting Few-Fermion Systems in One Dimension

The paper examines a trapped one-dimensional system of multicomponent spinless fermions that interact with a zero-range two-body potential. We show that when the repulsion between particles is very large the system can be approached analytically. To illustrate this analytical approach we consider a simple system of three distinguishable particles, which can be addressed experimentally. For this system we show that for infinite repulsion the energy spectrum is sixfold degenerate. We also show that this degeneracy is partially lifted for finitely large repulsion for which we find and describe corresponding wave functions.

Zero-range-potential model for strong-field molecular processes: Dynamic polarizability and photodetachment cross section

We develop a general theoretical framework for the analytical description of the interaction of a model diatomic molecular system with an intense, arbitrarily polarized monochromatic laser field. The model molecule comprises an electron in the field of two zero-range potentials separated by the internuclear distance. This model has an exact analytical solution within the theoretical framework of the quasistationary quasienergy (Floquet) approach. In addition to the development of this general framework, we also present a detailed analysis of the weak-field limit, within which we obtain both the frequency-dependent polarizability and the angle-resolved photodetachment cross section for the model system. These fundamental properties are analyzed for both homonuclear and heteronuclear molecular systems in a linearly polarized laser field, for both ground and excited electronic states, and for arbitrary orientation of the molecular axis relative to the polarization vector of the laser field. The analytical expressions for the polarizability and angle-resolved photoelectron spectra exhibit characteristic double-slit interference patterns, allowing one to study their dependence on the parameters of the problem beyond the level of the Born approximation.

Peculiarities of the tunneling ionization of atoms and ions in a magnetic field

We consider the tunneling ionization of an electron, bound by a zero-range potential and a constant magnetic field, under the influence of a low-frequency monochromatic laser beam. An integral equation for the complex quasienergy is derived based upon the exact solution of the Schr\"odinger equation and the Green's function earlier obtained by Rylyuk and Ortner [Phys. Rev. A 67, 013414 (2003)] for an electron moving in an arbitrary electromagnetic wave and a constant magnetic field. We show that accounting for an electron spin leads to drastic changes in ionization probability. For the case of a nonzero angle $\ensuremath{\theta}\ensuremath{\ne}0$ between an electromagnetic wave direction and the magnetic field the ionization rate is shown to increase with the magnetic field when the electron spin is taken into account. The expressions for the level shift and width and for the diamagnetic and the paramagnetic susceptibilities of the electron in a zero-range force field and a constant magnetic field are found. In the case of ionization of neutral atoms and positive ions, we also take into consideration the Coulomb interaction of the emerging electron with the atomic or ionic core.

Many-body properties of quasi–one-dimensional boson gas across a narrow CIR

We study strong interaction effects in a one-dimensional (1D) boson gas across a narrow confinement-induced resonance (CIR). In contrast to the zero-range potential, the 1D two-body interaction in the narrow CIR can be written as a polynomial of derivative δ-function interaction on many-body level. Using the asymptotic Bethe ansatz, we find that the low-energy physics of this many-body problem is described by the Tomonaga-Luttinger liquid where the Luttinger parameters are essentially modified by an effective finite-range parameter v. This parameter drastically alters quantum criticality and universal thermodynamics of the gas. In particular, it drives the Tonks-Girardeau (TG) gas from non-mutual Fermi statistics to mutual statistics or to a more exclusive super-TG gas. This novel feature is further discussed in terms of the breathing mode which is experimentally measurable.

Static-field-induced states

Considering an electron interacting with an atomic or molecular potential and an external static electric field, one usually focuses on narrow resonances at negative energies originating from the bound states in the absence of the field; we call them tunneling states (TSs). Meanwhile, there also exist relatively broad resonances at positive energies having no counterparts in the absence of the field; we call them static-field-induced states (SFISs). In this paper, the recently developed weak-field asymptotic theory of TSs [O. I. Tolstikhin et al., Phys. Rev. A 84, 053423 (2011)] is extended to SFISs. An asymptotic quantization condition defining the energies of SFISs in an arbitrary potential in the three-dimensional case is derived. The parabolic scattering amplitudes appearing in this quantization condition are defined. The theory is illustrated by calculations for the zero-range and Coulomb potentials. The SFISs in these potentials are found and their energies are shown to be in good agreement with the asymptotic results over a wide interval of the values of the field.

Zero-range potential for particles interacting via Coulomb potential

The zero-range potential is constructed for a system of two particles interacting via the Coulomb potential. The singular part of the asymptote of the wavefunction at the origin, which is caused by the common effect of the zero-range potential singularity and of the Coulomb potential, is explicitly calculated by using the Lippmann–Schwinger type integral equation. The singular pseudo potential is constructed from the requirement that it enforces the solution to the Coulomb Schrödinger equation to possess the calculated asymptotic behavior at the origin. This pseudo potential is then used for constructing a model of the imaginary absorbing potential for the positron–electron system. This potential allows us to treat the annihilation process in positron–electron collisions on the basis of the non-relativistic Schrödinger equation. The functional form of the pseudo potential constructed in this paper is analogous to the well known Fermi–Breit–Huang pseudo potential. The generalization of the optical theorem in the case of the imaginary absorbing potential in the presence of the Coulomb force is given in terms of the partial wave series.

H− − H Collision Induced Radiative Transitions

Exchange interaction leads to the formation of gerade and ungerade states of temporary molecules (quasimolecules) formed during the H− +H slow collisions. The work deals with the radiation produced by optical transitions between those states. The main characteristics involved in the description of optical transitions in quasimolecules, i.e., energy terms, an optical dipole transition moments, have been calculated in the frame of zero-range potentials model. The main feature of calculations is that the results can be expressed analytically in closed forms via the Lambert W function.

Interaction of an attosecond pulse with negative atomic and molecular ions

We report on the results of calculation of the probabilities of rescattering of ultrashort (attosecond) electromagnetic field pulses at one- and two-center negative ions in the s state in the zero-range potentials approximation; the dependence of the rescattering spectra on the number of field oscillations per pulse is investigated.

Asymptotics of the eigenvalues of a discrete Schrödinger operator with zero-range potential

We consider a family of discrete Schrödinger operators , . These operators are associated with the Hamiltonian of a system of two identical quantum particles (bosons) moving on the -dimensional lattice , , and interacting by means of a pairwise zero-range (contact) attractive potential . It is proved that for any there is a number which is a threshold value of the coupling constant; for the operator , , has a unique eigenvalue placed to the left of the essential spectrum. The asymptotic behaviour of is found as and as and also as for every value of the quasi-momentum belonging to the manifold , where .

Asymptotic behavior of an eigenvalue of the two-particle discrete Schrödinger operator

We consider a two-particle discrete Schrodinger operator corresponding to a system of two identical particles on a lattice interacting via an attractive pairwise zero-range potential. We show that there is a unique eigenvalue below the bottom of the essential spectrum for all values of the coupling constant and two-particle quasimomentum. We obtain a convergent expansion for the eigenvalue.

Asymptotic Bethe-ansatz solution for one-dimensional SU(2) spinor bosons with finite-range Gaussian interactions

We propose a one-dimensional model of spinor bosons with SU(2) symmetry and a two-body finite-range Gaussian interaction potential. We show that the model is exactly solvable when the width of the interaction potential is much smaller than the interparticle separation. This model is then solved via the asymptotic Bethe-ansatz technique. The ferromagnetic ground-state energy and chemical potential are derived analytically. We also investigate the effects of a finite-range potential on the density profiles through the local-density approximation. Finite-range potentials are more likely to lead to quasi-Bose-Einstein condensation than zero-range potentials.

Thermodynamics of the two-component Fermi gas with unequal masses at unitarity

We consider mass-imbalanced two-component Fermi gases for which the unequal-mass atoms interact via a zero-range model potential with a diverging s-wave scattering length $a_s$, i.e., with $1/a_s=0$. The high temperature thermodynamics of the harmonically trapped and homogeneous systems are examined using a virial expansion approach up to third order in the fugacity. We find that the universal part of the third-order virial coefficient associated with two light atoms and one heavy atom is negative, while that associated with two heavy and one light atom changes sign from negative to positive as the mass ratio $\kappa$ increases, and diverges when Efimov physics sets in at $\kappa=13.61$. By examining the Helmholtz free energy, we find that the equilibrium polarization of the trapped and homogeneous systems is 0 for $\kappa=1$, but finite for $\kappa \ne 1$ (with a majority of heavy particles). Compared to the equilibrium polarization of the non-interacting system, the equilibrium polarization at unitarity is increased for the trapped system and decreased for the homogeneous system. We find that unequal-mass Fermi gases are stable for all polarizations.

Persistent currents in dipolar Bose-Einstein condensates confined in annular potentials

We consider a dipolar Bose-Einstein condensate confined in an annular potential, with all the dipoles being aligned along some arbitrary direction. In addition to the dipole-dipole interaction, we also assume a zero-range hard-core potential. We investigate the stability of the system against collapse, as well as the stability of persistent currents as a function of the orientation of the dipoles and of the strength of the hard-core interaction.

On the number of eigenvalues of a model operator related to a system of three particles on lattices

We consider a quantum mechanical system on a lattice in which three particles, two of them being identical, interact through a zero-range potential. We admit a very general form for the ‘kinetic’ part H0γ of the Hamiltonian, which contains a parameter γ to distinguish the two identical particles from the third one (in the continuum case this parameter would be the inverse of the mass). We prove that there is a value γ* of the parameter such that only for γ < γ* the Efimov effect (infinite number of bound states if the two-body interactions have a resonance) is absent for the sector of the Hilbert space which contains functions which are antisymmetric with respect to the two identical particles, while it is present for all values of γ on the symmetric sector. We comment briefly on the relation of this result with previous investigations on the Thomas effect. We also establish the following asymptotics for the number N(z) of eigenvalues z below Emin, the lower limit of the essential spectrum of H0. In the symmetric subspace whereas in the antisymmetric subspace where , are written explicitly as a function of the integral kernel of operators acting on ( is the unit sphere in

A solvable model for homopolymers and self-similarity near the critical point

We consider a model for the distribution of a long homopolymer with an attractive zero-range potential at the origin in (polymer pinning and de-pinning). The distribution can be obtained as a limit of Gibbs distributions corresponding to properly normalized potentials concentrated in small neighborhoods of the origin as the size of the neighborhoods tends to zero. The distribution depends on the length T of the polymer and a parameter γ that corresponds, roughly speaking, to the difference between the inverse temperature in our model and the critical value of the inverse temperature. At the critical point γ cr = 0 the transition occurs from the globular phase (positive recurrent behavior of the polymer, γ &gt; 0) to the extended phase (Brownian type behavior, γ &lt; 0). The main result of the paper is a detailed analysis of the behavior of the polymer when γ is near γ cr . Our approach is based on analyzing the semigroups generated by the self-adjoint extensions ℒ γ of the Laplacian on parametrized by γ , which are related to the distribution of the polymer. The main technical tool of the paper is the explicit formula for the resolvent of the operator ℒ γ .

Ground states of the Schrödinger-Maxwell system with dirac mass: Existence and asymptotics

We study a non-relativistic charged quantum particle moving in a bounded open set $\Omega\subset\R^3$ with smooth boundary under the action of a zero-range potential. In the electrostatic case the standing wave solutions take the form $\psi(t,x)=u(x)e^{-i\omega t}$ where $u$ formally satisfies $-\Delta u+\alpha\varphi u-\beta\delta_{x_0} u=\omega u$ and the electric potential $\varphi$ is given by $-\Delta\varphi = u^2$. We introduce the definition of ground state. We show the existence of such solutions for each $\beta>0$ and the compactness as $\beta\to 0$.

Perturbation theory analysis of attosecond photoionization

Ionization of an atom by a few-cycle attosecond xuv pulse is analyzed using perturbation theory (PT), keeping terms in the transition amplitude up to second order in the pulse electric field. Within the PT approach, we present an ab initio parametrization of the ionized electron angular distribution (AD) using rotational invariance and symmetry arguments. This parametrization gives analytically the dependence of the AD on the carrier envelope phase (CEP), the polarization of the pulse, and on the ionized electron momentum direction, $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{p}}$. For the general case of an elliptically polarized pulse, we show that interference of the first- and second-order transition amplitudes causes a CEP-dependent asymmetry (with respect to $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{p}}\ensuremath{\rightarrow}\ensuremath{-}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{p}}$) and both elliptic and circular dichroism effects. All of these effects are maximal in the polarization plane and depend not only on the CEP but also on the phase of dynamical atomic parameters that enter our parametrization of the AD. Within the single active electron model of an atom, for an initial $s$ or $p$ state we define all dynamical parameters in terms of radial matrix elements (analytic expressions for which are given for the Coulomb and zero-range potentials). For ionization of the H atom by linearly polarized pulses, our PT results are in excellent agreement with results of numerical solutions of the time-dependent Schr\"odinger equation of Peng et al. [New J. Phys. 10, 025030 (2008)]. Also, our numerical results show that the asymmetries and dichroism effects at low electron energies have a different physical origin from those at high electron energies. Moreover, our results for Gaussian and cosine-squared pulse shapes are in good qualitative agreement. Finally, we show that our analytic formulas may prove useful for determining few-cycle extreme ultraviolet (xuv) pulse characteristics, such as the CEP and the polarization.