An expansion of continuum wave functions in terms of resonant states

The poles and residues of the complete outgoing Green function in the complex momentum plane are used to obtain, in the case of finite range potentials, an eigenfunction expansion of the continuum wave solution. It is found that in the region r<a the wave solution may be expressed as an infinite sum of discrete terms involving the bound, antibound and resonant states of the problem. At the boundary radius r = a a different expansion is obtained. In this case, in order to get an infinite discrete sum, one has to introduce two subtraction terms. Otherwise the expansion is given by a finite sum of discrete terms and an     

Test of the Q-space approximation in nuclear response theory

The electric monopole states are solved in TDA with a separable interaction taking into account the continuum. These results are compared with the Q-space approximation, in which the configuration space is discretized by using the harmonic oscillator basis, and the continuum is taken into account via the escape width. The Q-space approximation is found to be very good.     

A strong-coupling expansion for fermion field theories and the large-N limit of the gross-neveu model

An expansion in the fermion propagator is formulated for the N-species Gross-Neveu model in the large-N limit. Different regularisation schemes may be adopted and we compare two. We find that a continuum momentum cut-off is easiest to work with and automatically avoids spurious fermionic states which afflict a naive lattice formulation. Chiral symmetry is broken at zeroth order and the resulting expansion is inverse powers of g²N simplifies considerably for large N. In this limit the strong-coupling expansion may be summed to all orders. Extrapolation techniques, like Padé approximants, are not needed. Using a momentum cut-off we recover all the exact results previously derived by summing weak-coupling expansions.     

Resonances of (N±1)-particle systems in a dilated Green's function approach

The inclusion of dilatation transformations in the framework of the one-electron Green's function of many-body theory is discussed with regard to the calculation of resonance energies and widths of states containingN± 1 particles. By this mathematical method it is possible to extend the applicability of highly successful standard techniques employing only square integrable basis sets to metastable states which are degenerate with true continuum states into which they disintegrate with characteristic lifetimes.     

A resonance of the electronic polaron appearing in the optical absorption of alkali halides

Excited states (scattering states) of free and bound electronic polarons in non metals are introduced and investigated in the continuum approximation. It is suggested that transitions to these states might lead to prominent resonances in the optical absorption at energies approximately twice the bandgap energy. A shift towards higher energies of the corresponding resonances in the energy loss function is calculated. Such resonances and the predicted shift are found in the experimental data for alkali halides; previously they have generally been attributed to plasma excitations. Limitations of the present model, due to the continuum approximation, (and related to the oscillator strength of the transitions) are discussed. The electronic polaron coupling constant a is calculated and tabulated for a number of alkali halides.     

Particle-hole calculation of the isobaric analog and isovector monopole resonances

The correlated proton-particle—neutron-hole spectrum is calculated for N > Z nuclei using a Skyrme type interaction and the response function method. The basis of the calculation is a complete one-particle—one-hole space with the continuum included. As a result the distribution of the isovector monopole strength in the analog nucleus is obtained. This distribution has a narrow peak which corresponds to the isobaric analog resonance and at higher energies a broad peak which is the isovector monopole resonance. The coupling between these two states is inherent in the calculation.     

A low-lying resonance in the spectrum of H

It was proved by Pekeris⁽¹⁾ that the singly excited states of H^- lie exactly at, or slightly above, the ground state of hydrogen. Using a theory of Fano,⁽²⁾ these fictitious states will have a configuration interaction with the H^- continuum. The strength of this configuration interaction is computed for the mixing of a 1s2p¹P⁰ state with the H^- continuum for different values of the fictitious binding energy of the 2p valence electron. In every case, the effect of the configuration interaction is to induce a rapid change of the phase shift of the continuum wave function by a quantity of π/2 over an energy range of a few times 0.01 eV, at an energy somewhat above the hydrogen ground state. The variation from π/2 to π is much slower. Such a swift change of the phase shift may be identified with the occurrence of a low-lying shape resonance.     

Resonance scattering with modified continuum states: (II). Application of the unified model with one particle in the continuum to the 12C + n system

Starting from the conventional shell model, different methods for the treatment of the continuum are given. Using the modified continuum states of part I together with the reaction operator formalism, the present approach unifies the numerical features of the R-matrix theory and the physical picture of the shell model. As it treats the bound states and the scattering states in the same basis we calculated both for the ¹²C + n system as a test case. It is found that the results resemble the experimental data quite well, including for the first time the negative-parity resonances.     

Analytic continuation of scattering data to the region of negative energies for systems that have one and two bound states

An exactly solvable potential model is used to study the possibility of deducing information about the features of bound states for the system under consideration (binding energies and asymptotic normalization coefficients) on the basis of data on continuum states. The present analysis is based on an analytic approximation and on the subsequent continuation of a partial-wave scattering function from the region of positive energies to the region of negative energies. Cases where the system has one or two bound states are studied. The α+d and α+¹²C systems are taken as physical examples. In the case of one bound state, the scattering function is a smooth function of energy, and the procedure of its analytic continuation for different polynomial approximations leads to close results, which are nearly coincident with exact values. In the case of two bound states, the scattering function has two poles—one in the region of positive energies and the other in the region of negative energies between the energies corresponding to the two bound states in question. Padéapproximants are used to reproduce these poles. The inclusion of these poles proves to be necessary for correctly describing the properties of the bound states.     

Resonant states and their uses

The paper shows, for a simple model, how the wave functions belonging to complex energy eigenvalues (resonance states) can be regarded as part of a complete set of states, containing also bound states and a continuum of (generally complex) wave functions. This forms the basis of an expansion, which is used to describe inelastic scattering in the plane-wave Born approximation, taking the same simple model for the target.     

A low-lying resonance in the spectrum of H II. The 2P state

The wave function containing 20 parameters and the corresponding energy of the 2¹P state of H^- were calculated. As is well known this state lies in the H^- continuum. The inclusion of this state in the expansion of the continuum wave function can drastically influence the cross-section of an electron scattered by a hydrogen atom in the energy region involved. This bound pseudo-state will, in any case, influence the continuum absorption coefficient, but it may be insufficient to guarantee a peak in this absorption coefficient or in any e^-H experiment. Anyway, a new 2¹P virtual state of H^- is obtained with an energy of approximately -0.92 Ry. The exact position of the resonance needed to account for the occurrence of the interstellar line at 6180 Å is -0.9079 Ry. Since this resonance must lie very near the position of the bound state, our results support the identification of the broad feature at 6180 Å as a transition in H^-.     

Techniques to Treat the Continuum Applied to Electromagnetic Transitions in 8Be

Bremsstrahlung emission in collisions between charged nuclei is equivalent to nuclear gamma decay between continuum states. The way the continuum spectrum can be treated is not unique, and efficiency and accuracy of cross section calculations depend on the chosen method. In this work we describe, relate, and compare three different methods in practical calculations of inelastic cross sections, that is, by (i) treating the initial and final states as pure continuum states on the real energy axis, (ii) discretizing the continuum states on the real energy axis with a box boundary condition, and (iii) complex rotation of the hamiltonian (complex scaling method). The electric quadrupole transitions, 2⁺ → 0⁺ and 4⁺ → 2⁺, in α + α scattering are taken as an illustration.     

Parity Projected QCD Sum Rule of the Nucleon with MEM

The nucleon and its negative-parity excited states are examined in a maximum entropy method analysis of QCD sum rules. We derive the parity projected nucleon sum rules with all known first order α _s corrections to the Wilson coefficients of the operator product expansion (OPE). As these corrections have turned out to be large, we suppress them by using a phase-rotated Gaussian kernel. This phase rotation strongly suppresses the continuum contribution and improves the convergence of the OPE. The resulting sum rule has the interesting feature that it is dominated by the term containing the chiral condensate of dimension 3. Analyzing this sum rule by the maximum entropy method, we are able to extract information of both the positive and negative parity states.     

New approach to solving the problem of composite-particle scattering on nuclei

An essentially new approach to solving the problem of elastic and inelastic scattering of a composite particle on stable nuclei is described. Within this approach, all channels of virtual breakup and stripping in the intermediate states are included in a nonlocal complex-valued interaction operator with the aid of the projection-operator technique.The three-particle continuum spectrum of the Hamiltonian for intermediate states in Q space is calculated within the orthogonalizing-pseudopotential method by introducing a pseudo-Hamiltonian, which is diagonalized in a full space in terms of a relevant oscillator basis. As was shown by a number of authors, the use of special quadratures makes it possible to reduce integration over the continuous spectrum of intermediate states to summation over a discretized continuum. On the basis of the formalism developed in this study, a closed Schrödinger equation with a nonlocal complex potential for partial waves is derived for describing elastic scattering of a composite particle by a target, and an explicit approximate formula for the amplitude of three-particle breakup is obtained on the same basis. This method has a number of obvious advantages over currently well-known approaches of the type of the discretized-continuum coupled-channel method, where solving the problem in question reduces to solving a cumbersome set of coupled equations.     

Parabolic sturmians approach to the three-body continuum Coulomb problem

The three-body continuum Coulomb problem is treated in terms of the generalized parabolic coordinates. Approximate solutions are expressed in the form of a Lippmann-Schwinger-type equation, where the Green’s function includes the leading term of the kinetic energy and the total potential energy, whereas the potential contains the non-orthogonal part of the kinetic energy operator. As a test of this approach, the integral equation for the (e ^?, e ^?, He⁺⁺) system has been solved numerically by using the parabolic Sturmian basis representation of the (approximate) potential. Convergence of the expansion coefficients of the solution has been obtained as the basis set used to describe the potential is enlarged.     

Resonances in electron-impact integral excitation cross sections of the magnesium atom

The resonance structure of the integral cross sections of excitation of the magnesium atom by low-energy electrons is analyzed in terms of the R-matrix method. The collision strengths are calculated in terms of the close-coupling approximation with consideration of 25 atomic states, including 13 physical target states and 12 pseudostates. The latter simulate the remaining bound and continuum states of the Mg atom that were not included in the close-coupling expansion explicitly. The positions and widths of the resonances found are determined, and the resonances are classified with respect to the mechanism of formation of short-lived states of the negative Mg ion. The results of calculations are compared with available experimental data.     

Quarkonia at Finite T: An Approach Based On QCD Sum Rules and the Maximum Entropy Method

Quarkonium spectral functions at finite temperature are studied, making use of a recently developed method of analyzing QCD sum rules by the maximum entropy method. This approach enables us to directly obtain the spectral function from the sum rules, without having to introduce any specific assumption about its functional form. QCD sum rules for heavy quarkonia incorporate finite temperature effects in form of changing values of the various gluonic condensates that appear in the operator product expansion. These changes depend on the energy density and pressure at finite temperature, which we extract from quenched lattice QCD calculations. As a result, it is found that the charmonium ground states of both S-wave and P-wave channels dissolve into the continuum already at temperatures around or slightly above the critical temperature T _c , while the bottomonium states are less influenced by temperature effects, surviving up to about 2.5 T _c or higher for S-wave and about 2.0 T _c for P-wave states.     

Continuum three-body states using the hyperspherical adiabatic basis set

In this paper we investigate the feasibility of employing the hyperspherical adiabatic (HA) basis set to describe continuum three-nucleon states. In particular, the HA expansion is compared with the simpler expansion on hyperspherical harmonics (HH). A practical numerical application is presented using the MT-III potential.     

Defects in the Sine-Gordon model: Statics

The one-dimensional Sine-Gordon model where the field couples linearly to localized defects is studied. This model describes the phase fluctuations of one-dimensional charge density waves if amplitude fluctuations are neglected. For isolated defects the nonlinear stationarity conditions can be solved analytically. Given these stationary solutions the fluctuation spectrum consisting of a bound state and the scattering states and the relevant correlation functions can be computed exactly. The stationary states for a system of two defects are presented. The free energies as a function of the separation of the defects is computed for the absolutely stable states and for the local minima of the free energy function. This allows us to consider the interaction of the defects induced by the Sine-Gordon field. Finally we compute the order parameter correlation function for a random distribution of defects for small concentrationn. Devising a cumulant expansion the correlation function is found exactly in the first order inn. Our result contains the combined effect of the defects on the stationary states and on the phonon Green's function. It has applicability beyond the present context.     

An expansion of continuum wave functions in terms of resonant states: (II). Solvable models

This paper applies the formalism developed in part I which provides a purely discrete expansion for a continuum wave solution of the Schrödinger equation in terms of resonant states along the interior region of a finite range interaction. We consider two exactly solvable models for several values of the distance and momenta on and off resonance. Our results are compared with a recent approach by Bang and collaborators which involves subtraction terms. It is found that along the internal region the subtraction terms are not in general negligible. Our work substantiates an expression for the continuum wave function near resonance. We also obtain an equation for time delay in terms of resonant states.