The 2007 study of Michel, Nazarewicz and Ploszajczak (Nucl. Phys. A 794, 29) demonstrated, within the Gamow shell model (GSM), that coupling to the particle continuum produces a near-threshold, non-perturbative rearrangement of nuclear wave functions with characteristic Wigner-threshold scaling of spectroscopic factors, and that the complex single-nucleon overlap functions required by reaction theory can be generated by single-particle states of a complex Woods-Saxon potential matched to the complex generalized separation energy. The paper's conclusions explicitly raised: (i) construction and realistic testing of a self-consistent complex average potential (Gamow-Hartree-Fock), including whether its imaginary part can be kept small enough to avoid mathematical complications; and (ii) the modification of the Berggren completeness relation needed when the one-body potential has a sizable imaginary part (multiple S-matrix poles, additional non-eigenfunction contributions, spectral singularities). Two related directions complete the program: (iii) systematic quantification of continuum-induced rearrangements beyond the light helium and oxygen chains (the paper's own demonstrations, with effective interactions, were explicitly illustrative); and (iv) integration of the complex-overlap prescription into reaction models (the paper called for a unification of structure and reaction aspects of weakly bound or unbound nuclei).

Background

Near the particle drip lines, atomic nuclei are weakly bound or unbound, so their low-lying states are open quantum systems coupled to the scattering continuum (the continuum of unbound particle states, of both resonant and non-resonant character). Two standard structure observables are sensitive to this coupling. The single-nucleon overlap function is the wave-function overlap between a parent nucleus and a daughter nucleus plus one nucleon; it is the basic microscopic input to direct transfer and knockout reactions. Its norm (with respect to a bi-orthogonal generalization for complex energies) is the spectroscopic factor (SF), which quantifies how much a state looks like a pure single-particle configuration attached to a core. Experimentally extracted SFs from transfer and knockout reactions are among the most debated observables in nuclear spectroscopy, so a controlled theory of overlap functions near threshold matters well beyond structure physics. The Gamow shell model (GSM), introduced by Michel, Nazarewicz, Ploszajczak and collaborators, is the configuration-interaction shell model generalized to this open regime: the usual discrete single-particle basis is replaced by the Berggren basis, built from the Berggren completeness relation. This relation, derived by Berggren in 1968 for real (Hermitian) finite-depth potentials, states that the identity can be decomposed as the sum over bound states and Gamow resonant states plus an integral over scattering states along a complex momentum contour. Gamow states are resonant single-particle states with complex energy Eres=E−iΓ/2E_\text{res}=E-i\Gamma/2 (energy minus half the decay width); they obey outgoing-wave boundary conditions and are not square-integrable, so they live outside the ordinary Hilbert space and require a bi-orthogonal (left/dual) basis. In this setting, Hermiticity of the Hamiltonian is given up, but flux conservation (unitarity) near threshold is respected exactly. Within GSM, the 2007 source paper studied one-neutron overlap integrals in neutron-rich helium and oxygen chains. It showed (a) that the near-threshold behavior of SFs follows the Wigner threshold law — the general result, first noted by Wigner in 1948, that near-threshold observables behave as a power of the channel energy with exponents fixed by the orbital angular momentum ℓ\ell; GSM reproduced this many-body Wigner cusp only when the non-resonant continuum was included explicitly; and (b) that when parent or daughter is unbound the overlap function is complex, and it can be accurately approximated by a single-particle resonant state of a complex Woods-Saxon potential adjusted to the complex generalized one-nucleon separation energy S~1n\tilde{S}_{1n}, the complex analogue of the usual binding-energy difference. The paper then identified what is missing to make this prescription fully microscopic and internally consistent. First, the complex average potential is phenomenological: optimally it should be generated self-consistently from the underlying interaction by a Gamow-Hartree-Fock (GHF) procedure — Hartree-Fock mean-field theory carried out in the Berggren ensemble — but no such self-consistent complex mean field had been implemented for realistic unbound nuclei. Second, the mathematical foundation becomes delicate when the one-body potential is strongly non-Hermitian: Berggren derived his completeness relation only for real potentials; for complex potentials, multiple SS-matrix poles can appear in the resonant-state expansion, requiring additional states that are not eigenfunctions of the one-body Hamiltonian, and spectral singularities — exceptional situations where an SS-matrix pole sits on the real momentum axis and the Gamow state degenerates into a non-square-integrable, self-dual object — can further obstruct the resolution of the identity. Third, the Wigner-threshold scalings had been demonstrated only for ℓ=1\ell=1 (heliums) and ℓ=2\ell=2 (oxygens) with effective interactions. Fourth, the complex-overlap prescription had yet to be consistently folded into reaction models. Subsequent methodological context: the GSM was later extended to the coupled-channel GSM (GSM-CC), which unifies structure and reaction descriptions in the same complex-energy framework, and the whole subject is surveyed in the 2008 review 'Shell model in the complex energy plane' and the 2021 monograph 'Gamow Shell Model' by Michel and Ploszajczak.

Problem Statement

Within the complex-energy (Berggren/Gamow) formulation of the nuclear shell model, establish a fully self-consistent microscopic description of single-nucleon overlap functions and spectroscopic factors in weakly bound and unbound nuclei, as called for by Michel, Nazarewicz and Ploszajczak (Nucl. Phys. A 794 (2007) 29). The common objective is to remove the remaining phenomenological and mathematical inconsistencies of the complex-overlap prescription; its linked components are: (i) construct, implement, and test in realistic cases a self-consistent complex average potential - a Gamow-Hartree-Fock mean field derived from the underlying interaction, in which the imaginary part generated by coupling to decay channels is retained and one establishes to what extent it can be kept small enough (or treated) so that the associated Gamow states and contour expansions remain mathematically well behaved; (ii) derive the Berggren completeness relation for non-Hermitian localized one-body potentials with a sizable imaginary part, specifying the additional non-eigenfunction contributions associated with multiple S-matrix poles and the treatment of spectral singularities, so that many-body expansions in such bases are rigorous; (iii) quantify the magnitude and assess the universality of continuum-induced near-threshold rearrangements - the Wigner-threshold scaling of spectroscopic factors - beyond the original ℓ=1\ell=1 helium and ℓ=2\ell=2 oxygen examples, in heavier nuclei, across a broader set of partial waves, and with varied interactions, including the consequences for the experimental extraction of spectroscopic factors; and (iv) integrate the complex-overlap prescription consistently into reaction models (transfer, knockout, capture), with a consistent treatment of the imaginary parts in reaction amplitudes and of the associated uncertainties, benchmarked against experimental data. A solution must advance both (i) and (ii); (iii) and (iv) are optional extensions and are not required for resolution.

References

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