Description of resonant states in the shell model

A technique for describing scattering states within the nuclear shell model is proposed. This technique is applied to scattering of nucleons by particles based on ab initio No-Core Shell Model calculations of ⁵He and ⁵Li nuclei with JISP16 NN interaction.     

Two-body T-matrix continued off the energy shell

A generalized parametrization scheme, which treats both half-shell and off-shell scattering on equal footing, is derived for the two-particle transition operator. The appropriate theory in respect of this is developed by using a wave function approach to off-shell scattering and the computational procedure to be followed is demonstrated by means of a model calculation.     

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     

Weak coupling expansions fully off the energy shell

This paper contains off-shell generalizations of the standard (on-shell) Born series. Both Fredholm and Volterra integral equations are derived for the half-off-shell wave functions. Fully off-shell matrix elements of the transition operator are then developed by iteration. Convergence and truncation problems are also discussed. An alternative approach via the explicit form of the T-operator is also indicated.     

On the mean value of the energy for resonant states

In this work we discuss possible definitions of the mean value of the energy for a resonant (Gamow) state. The mathematical and physical aspects of the formalism are reviewed. The concept of rigged Hilbert space is used as a supportive tool in dealing with Gamow-resonances.     

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.     

Study of the structures of four-quark states in terms of the Born-Oppenheimer approximation

In this work,we use the Born-Oppenheimer approximation,where the potential between atoms can be approximated as a function of distance between the two nuclei,to study the four-quark bound states.By this approximation,Heitler and London calculated the spectrum of the hydrogen molecule,which includes two protons(heavy)and two electrons(light).Generally,the observed exotic mesons Zb(10610),Zb(10650),Zc(3900)and Zc(4020)(Zc(4025))may be molecular states made of two physical mesons and/or diquark-anti-diquark structures.Analogous to the Heitler-London method for calculating the mass of the hydrogen molecule,we investigate whether there exist energy minima for these two structures.Contrary to the hydrogen molecule case where only the spin-triplet possesses an energy minimum,there exist minima for both of these states.This implies that both molecule and tetraquark states can be stable objects.Since they have the same quantum numbers,however,the two states may mix to result in the physical states.A consequence would be that partner exotic states co-existing with Zb(10610),Zb(10650),Zc(3900)and Zc(4020)(Zc(4025))are predicted and should be experimentally observed.     

Non-sequential effects in the decay of resonant states

Fano and Prats' treatment of continuum-continuum interaction is used to study the time-dependent behaviour of a system initially in a discrete state coupled to a continuum which is itself coupled to another continuum. This level scheme represents a limiting case in a model which is currently used in the theory of molecular processes, the opposite limit being the sequential scheme which leads to chain-like rate equations. A very different behaviour is obtained in the two limits. In contrast with the results of the sequential model the continuum-continuum coupling produces a change (a decrease) in the decay rate of the discrete state. The two continua are populated in parallel. The calculations provide an explicit example of the influence of higher order terms on a rate constant, a problem which has received attention recently in relation to the evaluation of non-radiative transition rates using approximate molecular wave functions.     

Probabilistic interpretation of resonant states

We provide probabilistic interpretation of resonant states. We do this by showing that the integral of the modulus square of resonance wave functions (i.e., the conventional norm) over a properly expanding spatial domain is independent of time, and therefore leads to probability conservation. This is in contrast with the conventional employment of a bi-orthogonal basis that precludes probabilistic interpretation, since wave functions of resonant states diverge exponentially in space. On the other hand, resonant states decay exponentially in time, because momentum leaks out of the central scattering area. This momentum leakage is also the reason for the spatial exponential divergence of resonant state. It is by combining the opposite temporal and spatial behaviours of resonant states that we arrive at our probabilistic interpretation of these states. The physical need to normalize resonant wave functions over an expanding spatial domain arises because particles leak out of the region which contains the potential range and escape to infinity, and one has to include them in the total count of particles.     

Off shell states in the dual model

We present a class of Lorentz invariant, gauge invariant off-shell amplitudes for the ordinary Veneziano and Ramond-Neveu-Schwarz dual models. They correspond to interactions with the string which are local in space-time.     

Relations between the wave solutions and the jost solutions off the energy shell for local central potentials

Using the Green function techniques we express the wave solutions of the radial inhomogeneous Schrödinger equation by means of the on-shell Jost and regular solutions. Making use of their boundary behaviour atr = andr = 0 we reexpress them alternatively in terms of the off-shell Jost and regular solutions. Relations among the different generalized (fully off the energy shell) Jost functions are derived and the radial matrix elements of the transition and reaction (reactance) operators are given in terms of these Jost functions. The relations reflect the principle of detailed balance.     

On the strong influence of inner shell resonances upon the outer shell photoionization of endohedral atoms

It is demonstrated by the example of the Xe atom stuffed inside the C₆₀ fullerene, i.e., the endohedral Xe@C₆₀, that the so-called confinement resonances in 4d subshell strongly affect the photoionization cross section of outer 5p and subvalent 5s electrons near the 4d ionization threshold. It is a surprise that these narrow inner 4d shell resonances are not smeared out in the outer shell photoionization cross section. On the contrary; the inner shell resonances affect the outer cross section by enhancing them enormously. Close to its own photoionization thresholds, 5p and 5s photoionization cross sections of Xe@C₆₀ are dominated by their own confinement resonances greatly affected by the amplification of the incoming radiation intensity due to polarization by it of the C₆₀ electron shell. Between 4d and 5p thresholds, the effect of 4d is becoming stronger while own resonances of 5p and 5s are becoming much less important.     

Doublet resonances of an electron in symmetric three-barrier resonant tunneling structures

The evolution and collapse of electron resonances and their spectral parameters in a symmetric three-barrier resonant tunneling structure (TBRTS) are studied theoretically. The resonance energy and width of the quasi-stationary states of an electron are analyzed. The quasi-stationary states are calculated by the transmission coefficient method, the method of the probability distribution function (the probability of finding an electron in a TBRTS), and the scattering cross section method.     

Hybrid magneto-electric states in resonant tunnelling structures

Double barrier resonant tunnelling structures with wide undoped quantum wells are used to study quantum ballistic transport in the presence of a magnetic field B. The structures are based on n⁻GaAs/(AlGa)As with well widths of 60 and 120 nm. At B=0, the wider well structure (120 nm) shows as many as 70 resonances in I(V). With B applied in the plane of the barriers (B·J) these resonances evolve into hybrid magneto-electric states. At sufficiently large B, the electron orbits no longer extend to the second barrier and tunnelling occurs into cycloidal interface states which are localised near the emitter barrier. A theoretical model for the observed resonances based on the quantisation of the hybrid and cycloidal orbits is presented. Ballistic path lengths of at least 400 nm are observed.     

Relativistic resonances as non-orthogonal states in Hilbert space

We analyze the energy-momentum properties of relativistic short-lived particles with the result that they are characterized by two 4-vectors: in addition to the familiar energy-momentum vector (timelike) there is an energy-momentum `spread vector' (spacelike). The wave functions in space and time for unstable particles are constructed. For the relativistic properties of unstable states we refer to Wigner's method of Poincaré group representations that are induced by representations of the space-time translation and rotation groups. If stable particles, unstable particles and resonances are treated as elementary objects that are not fundamentally different one has to take into account that they will not generally be orthogonal to each other in their state space. The scalar product between a stable and an unstable state with otherwise identical properties is calculated in a particular Lorentz frame. The spin of an unstable particle is not infinitely sharp but has a `spin spread' giving rise to `spin neighbors'. This opens the possibility of a non-zero scalar product between states with unequal spin. - A first practical application of non-orthogonal states is seen in diffraction dissociation reactions whose large cross-sections are attributed to interference of states that are `partially identical'. Received: 28 June 2002 / Revised version: 13 November 2002 / Published online: 14 March 2003 RID="a" ID="a" e-mail: walter.blum@cern.ch RID="b" ID="b" e-mail: hns@mppmu.mpg.de     

Bound states and resonant states of three self-gravitating Bose-Einstein condensates

The effect of Yukawa potentials on the bound states and resonance states for systems with three self-gravitating bosons and fermions has been investigated using highly correlated basis functions. The complex coordinate rotation method is used to extract resonance parameters. We have investigated the Borromean binding for such systems. The ground and excited state energies, resonance energy and width for the lowest two resonances are reported for different screening parameters along with the critical range for Borromean binding.