Completing the deformed Berggren-basis program: non-axial momentum-space construction of the single-particle continuum ensemble and many-body Gamow-shell-model applications to deformed halo nuclei

The contour deformation method (CDM) constructs the single-particle Berggren ensemble — bound states, resonant (Gamow) states, and the nonresonant continuum on a complex momentum contour — by solving the momentum-space Schrödinger equation along a contour in the complex-$k$ plane. Hagen and Vaagen (Phys. Rev. C 73, 034321 (2006)) generalized this construction to deformed mean fields but demonstrated it only for axially symmetric Gaussian potentials without spin and tensor components, on a fictitious 'deformed $^5$He', and left open the extensions to more general (non-axial) deformation and to many-body Berggren bases for truly deformed halo nuclei such as $^8$Li, $^9$Be, and $^{11}$Be. Later literature has filled parts of this program at the single-particle level and in alternative frameworks: axially deformed resonances are now computed by complex scaling, Green's-function, and relativistic complex-momentum-representation methods (the latter including spin-orbit through the Dirac structure), and deformed halo nuclei have been described by continuum-coupled particle-plus-rotor models (a valence particle coupled to a rotating deformed core) formulated in the Berggren ensemble. However, no published work constructs the Berggren ensemble for non-axially symmetric (triaxial) deformed fields, and no many-body configuration-interaction calculation has been performed on a basis of deformed single-particle Berggren states; the Gamow shell model remains restricted to spherical bases. The open problem is the completion of this program: the generalized construction of the deformed single-particle Berggren ensemble beyond axial symmetry and its use in many-body Gamow-shell-model-type calculations of genuinely deformed halo nuclei with the nonresonant continuum retained.

Background

Nuclei near the neutron drip line — the boundary of the chart of nuclei beyond which an additional neutron is no longer bound — are weakly bound open quantum systems: their last neutrons occupy states that are barely bound or unbound. An unbound single-particle state is a resonance, described by a Gamow state with complex energy $E_r-i\Gamma/2$, where $\Gamma$ is the decay width and the wave function satisfies purely outgoing boundary conditions. Halo nuclei such as $^{11}$Be are the extreme case, with one or two valence neutrons spreading into an extended density tail. Since such nuclei are often also deformed — their average (mean) field is not spherical — a faithful description must combine deformation with the continuum.

The Berggren ensemble provides the mathematical tool: a generalized completeness relation in which a finite number of bound and resonant Gamow states, together with an integral along a complex contour over the nonresonant scattering continuum, forms a complete single-particle basis. The contour deformation method constructs this ensemble in momentum space: the single-particle Schrödinger equation becomes an integral equation which, when solved along a contour $L^+$ in the complex momentum plane, yields bound, resonant, and continuum states directly by matrix diagonalization, with boundary conditions automatic because no differential propagation is involved. The Gamow shell model (GSM) then builds many-body Slater determinants from these Berggren orbitals, giving a configuration-interaction description in which bound states, resonances, and continuum coupling are treated on the same footing; the complex-energy shell model is well developed for spherical systems.

For deformed fields the situation in the mid-2000s was far less complete. Ferreira, Maglione, and Liotta had obtained resonances in axially deformed Woods-Saxon potentials (the standard phenomenological nuclear mean-field potential) by solving coupled radial equations with outgoing asymptotics, but such coupled-channel methods do not give access to the nonresonant continuum, so a complete deformed basis and real observables were out of reach. In this context Hagen and Vaagen (2006) generalized the momentum-space CDM to deformed fields: expanding an axially symmetric deformed Gaussian potential — deliberately chosen without spin and tensor components — in angular harmonics, they reduced the problem to coupled one-dimensional integral equations on $L^+$ and obtained a deformed single-particle Berggren ensemble, demonstrated on a fictitious 'deformed $^5$He'. They stated explicitly that the method may be extended to more general deformation and applied to truly deformed halo nuclei, mentioning $^8$Li, $^9$Be, and $^{11}$Be, and that the Berggren states may be used in the construction of a many-body Berggren basis.

Two decades of subsequent work has filled parts of this program at the single-particle level and in alternative frameworks, but not its core: all existing complex-momentum constructions for deformed fields remain restricted to axial symmetry, where the projection $\Omega$ of the angular momentum on the symmetry axis is conserved, and no many-body configuration-interaction calculation has ever been performed on a basis of deformed single-particle Berggren states.

Problem Statement

The problem, as posed by the source paper and still open after two decades, is to complete the deformed Berggren program in momentum space: (i) generalize the contour-deformation solution of the momentum-space Schrödinger equation to deformed mean fields without axial symmetry — triaxial or fully asymmetric shapes, for which no projection quantum number is conserved and many multipole harmonics couple simultaneously; (ii) establish that the resulting states form a genuine Berggren ensemble, i.e. bound states, resonant Gamow states, and the nonresonant continuum on the chosen complex contour together satisfy the generalized completeness relation, so that observables computed with the nonresonant continuum retained are stable against basis truncation and contour discretization; and (iii) construct many-body Berggren bases from these deformed single-particle states and apply them in Gamow-shell-model-type calculations to genuinely deformed halo nuclei such as $^8$Li, $^9$Be, and $^{11}$Be, extracting spectra, decay widths, and structure observables with continuum effects quantified. The three requirements (i)–(iii) are cumulative: the program is completed, and the task is resolved, only when all three are delivered — the non-axial single-particle construction, its completeness and stability under basis truncation and contour discretization, and the many-body Gamow-shell-model-type application with the nonresonant continuum retained; a result that supplies only some of them does not resolve the task.

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