Analytic Continuation of Eigenvalues of a Quartic Oscillator

We consider the Schrödinger operator on the real line with even quartic potential x ⁴ + α x ² and study analytic continuation of eigenvalues, as functions of parameter α. We prove several properties of this analytic continuation conjectured by Bender, Wu, Loeffel and Martin. 1. All eigenvalues are given by branches of two multi-valued analytic functions, one for even eigenfunctions and one for odd ones. 2. The only singularities of these multi-valued functions in the complex α-plane are algebraic ramification points, and there are only finitely many singularities over each compact subset of the α-plane.     

Bound states of three and four resonantly interacting particles

We present an exact diagrammatic approach for the problem of dimer-dimer scattering in 3D for dimers being a resonance bound state of two fermions in a spin-singlet state, with corresponding scattering length a _F . Applying this approach to the calculation of the dimmer-dimer scattering length a _B , we recover exactly the already known result a _B = 0.6 a _F . We use the developed approach to obtain new results in 2D for fermions and bosons. Namely, we calculate bound state energies for three bbb and four bbbb resonantly interacting bosons in 2D. For the case of resonance interaction between fermions and bosons, we exactly calculate bound state energies of the following complexes: two bosons plus one fermion bbf, two bosons plus two fermions bf↑bf↓, and three bosons plus one fermion bbbf.     

Four-particle problem using Feynman diagrams

We present an exact diagrammatic approach for the problem of dimer-dimer scattering in 3D for dimers being a resonant bound state of two fermions in a spin singlet state, with corresponding scattering length a. We recover exactly the previously known result a _B = 0.60a, where a _B is the dimer-dimer scattering length. A detailed discussion of how one can “sum all the diagrams” in this case is presented. Applications to the study of 4-particle bound states of various complexes in 2D are briefly presented.     

Complex singularities of the fluid velocity autocorrelation function

There are intensive debates regarding the nature of supercritical fluids: if their evolution from liquid-like to gas-like behavior is a continuous multistage process or there is a sharp well-defined crossover. Velocity auto-correlation function Z is the established detector of evolution of fluid particles dynamics. Usually, complex singularities of correlation functions give more information. For this reason, we investigate Z in complex plane of frequencies using numerical analytic continuation. We have found that naive picture with few isolated poles fails describing Z(ω) of one-component Lennard-Jones (LJ) fluid. Instead, we see the singularity manifold forming branch cuts extending approximately parallel to the real frequency axis. That suggests LJ velocity autocorrelation function is a multivalued function of complex frequency. The branch cuts are separated from the real axis by the well-defined “gap” whose width corresponds to an important time scale of a fluid characterizing crossover of system dynamics from kinetic to hydrodynamic regime. Our working hypothesis is that the branch cut origin is related to competition between one-particle dynamics and hydrodynamics. The observed analytic structure of Z is very stable under changes in the temperature; it survives at temperatures two orders of magnitude higher than the critical one.     

SS-HORSE Method for Analysis of Resonances: Charged-Particle Scattering

The method for analyzing resonance states based on the harmoni©oscillator representation of scattering equations (HORSE) and on analytic properties of partial-wave scattering amplitudes which has been proposed earlier, is generalized to the case of charged-particle scattering. The method in question is tested by applying it to the model problem of pα scattering and can be used to study resonance states on the basis of microscopic calculations within various versions of the shell model.     

Analytic continuation of the effective-range expansion as a method for determining the features of bound states: Application to the 6Li nucleus

The scattering function (effective-range function) for the two-channel elastic scattering of charged particles is used to analyze dα scattering at low energies. In order to construct this function, use is made of various sets of phase shifts and mixing parameter, both those that were obtained by solving Faddeev equations in the three-body (n, p, α) model and those that were deduced from available phaseshift analyses. By means of an analytic continuation of the scattering function to the point of the pole corresponding to the bound state of the ⁶Li nucleus, the values of the vertex constants and asymptotic normalization coefficients are found for the process ⁶Li → α + d. Possible means for refining these results are discussed.     

Scattering of <Emphasis Type="Italic">π</Emphasis><Superscript>±</Superscript> mesons on a <Superscript>9</Superscript>Be nucleus in the Δ<Subscript>33</Subscript>-resonance region

Within the Glauber diffraction theory of multiple scattering, the differential cross sections for the elastic and inelastic scattering of π^± mesons are calculated for energies in the range between 130 and 260 MeV. This is the region where the broad Δ₃₃ resonance in the π^±N system occurs, the maximum corresponding to this resonance being at approximately 165 MeV. The wave function for the ⁹Be nucleus was chosen on the basis of the 2αN multicluster model. The sensitivity of the resulting differential cross sections to the target-nucleus wave functions computed with various intercluster-interaction potentials, to the contributions of wave-function components, and to various scattering multiplicities in the Glauber operator Ω is analyzed. A comparison with experimental data and with the results of other calculations is performed, and conclusions concerning the quality of the wave functions used and advantages of the present approach are drawn.     

Investigation of the neutron shell structure of the even-even isotopes <Superscript>40–56</Superscript>Ca within the dispersive optical model

Within the method of matching experimental data obtained in the neutron-stripping and neutron-pickup reactions on ^{40,42,44,46,48}Ca isotopes, the single-particle energies and probabilities that neutron states are filled are obtained for the even-even calcium isotopes. These data are analyzed within the dispersive optical model, and good agreement between the calculated and experimental values of the energies of states is obtained. The dispersive optical potential is extrapolated to the region of the unstable ^50,52,54,56Ca nuclei. The calculated single-particle energies of bound states in these isotopes are compared with the results of the calculations within the multiparticle shell model, the latter predicting a new magic number N = 34 for Z = 20 nuclei.     

The resonance-like structure observed in the <Superscript>25</Superscript>Mg(p, γ)<Superscript>26</Superscript>Al reaction

The gamma decay of resonance-like structure (RLS) observed in the ²⁵Mg(p, γ)²⁶Al reaction in the region of excitation energies of 7–9 MeV is studied, and the excitation function of this reaction is measured. The resonance transition strengths of states in the 1.4–2.0 MeV range of accelerated proton energies are determined. The resulting strength distributions of M1 transitions between the resonance states and the low-lying bound states in ²⁶Al are of a resonance nature. The position of the center of gravity of magnetic dipole resonance for the ground state in ²⁶Al is 7.92 MeV, and the total MDR transition strength is 5.7 MeV × μ _N ² .     

The Laughlin liquid in an external potential

We study natural perturbations of the Laughlin state arising from the effects of trapping and disorder. These are N-particle wave functions that have the form of a product of Laughlin states and analytic functions of the N variables. We derive an upper bound to the ground state energy in a confining external potential, matching exactly a recently derived lower bound in the large N limit. Irrespective of the shape of the confining potential, this sharp upper bound can be achieved through a modification of the Laughlin function by suitably arranged quasi-holes.     

Alpha-cluster states in <Superscript>18</Superscript>O

The excitation function of elastic α-particle scattering on ¹⁴C has been measured in the laboratory energy range 16.3–19.2 MeV using a backscattering technique with a thick target. These data were analyzed together with the old low-energy data of G.L. Morgan et al. in the framework of the R-matrix formalism. Spin-parity assignments were made for 32 states in ¹⁸O in the excitation range 9–20 MeV. The estimates of the widths of the states are also presented. The 0⁺ and 0^?α-cluster bands appeared to be well separated by 5.6 MeV (as in ¹⁶O and ²⁰Ne). We have not found a confirmation of existence of the negative-parity molecular states proposed by M. Gai et al. We observed an effect of a doubling of α-cluster levels in ¹⁸O similar to that found in ²²Ne.     

Analytical approach to constructing effective hadron-hadron and hadron-nucleus interaction operators in low-momentum space of states

The approach to constructing effective hadron-hadron and hadron-nucleus potentials in the low-momentum space of the states, based on the use of analytic properties of scattering amplitudes and solution of the inverse quantum scattering problem for determining short-range interaction components is developed. The approach is applied to constructing effective potentials in S-wave channels of NN and n³H scattering.     

Approximate path integral solution for a Dirac particle in a deformed Hulthén potential

The problem of a Dirac particle moving in a deformed Hulthén potential is solved in the framework of the path integral formalism. With the help of the Biedenharn transformation, the construction of a closed form for the Green’s function of the second-order Dirac equation is done by using a proper approximation to the centrifugal term and the Green’s function of the linear Dirac equation is calculated. The energy spectrum for the bound states is obtained from the poles of the Green’s function. A Dirac particle in the standard Hulthén potential (q = 1) and a Dirac hydrogen-like ion (q = 1 and a → ∞) are considered as particular cases.     

Investigation of the reaction mechanism for the four-particle photodisintegration of a carbon nucleus

The four-particle photodisintegration of a carbon nucleus in the reactions ¹²C(γ, p)³H2α and ¹²C(γ, n)³H2α is investigated by a method that employs a diffusion chamber in a magnetic field. It is shown that these reactions proceed according a sequential-type scheme: excited states of ¹¹B and ¹¹C nuclei decay to weakly excited states of ⁸Be, ⁷Li, and ⁷Be nuclei. It is concluded that nucleons are knocked out from the s shell. In the excitation curve for the 2α system in the reaction ¹²C(γ, p)³H2α, a resonance is found between the maxima corresponding to the ground and the first excited state of the ⁸Be nucleus, and this resonance is identified as a ghost anomaly. The branching fractions of the decay modes are determined. The angular distributions of nucleons in the reaction c.m. frame are measured. The energy dependence of the asymmetry coefficient for the angular distributions is obtained. A fast increase in this coefficient is observed in the energy range 38–40 MeV. It is concluded that the asymmetry coefficient depends on the excitation energy of the final nucleus in the region of intermediate photon energies.     

Resonant parameters of <Superscript>5</Superscript>He and <Superscript>5</Superscript>Li states and nucleon-α scattering

A new method of analysis of resonant parameters in the framework of the J-matrix inverse scattering formalism is proposed. The method is applied to analysis of Nα scattering in various partial waves. The obtained 1/2^? and 3/2^? resonance energies and widths in ⁵He and ⁵Li nuclei are in good agreement with the results of other approaches. The eigenenergies entering the J-matrix phase shift parameterization are shown to correlate well with the respective results of no-core shell model calculations, in particular, in the case of non-resonant s-wave scattering.     

Clusters and covalently bound nuclear molecules

Nuclear clustering based on α particles and strongly bound substructures with N=Z has been studied for many decades. Of particular interest are excited states close to the decay thresholds into substructures, as described by the Ikeda diagram. This diagram can be extended to neutron-rich nuclei; in these cases strongly deformed isomeric states consisting of clusters and loosely bound neutrons will appear. A possible approach to describe these states is to use explicitly molecular concepts, with neutrons in covalent binding orbits. Examples for molecular structure in beryllium isotopes and in other neutron-rich light nuclei (carbon and neon) are discussed.