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As a continuation of our previous work on the conservation and breaking of the pseudospin symmetry (PSS) in resonant states [Phys. Lett. B 847, 138320 (2023)], in this work, the PSS in nuclear single-particle bound and resonant states are investigated uniformly within a relativistic framework by exploring the poles of the Green's function in spherical Woods-Saxon potentials. As the potential depths increase from zero to finite depths, the PS partners evolve from resonant states to bound states. In this progress, the PSS is broken gradually with energy, width, and density splittings. Specially, the energy and width splittings for the resonant and bound states are directly determined by the ratio of the pseudo spin-orbit potentials between the PS partners. Obvious threshold effect is observed for the energy splitting at a critical potential depth, with which one PS partner has become a quasi-bound state inside the centrifugal barrier while the other one is still a high-energy resonant state outside the centrifugal barrier. The differences in the density distributions of the lower component between the PS partners are manifested in the phase shift for the resonant states and amplitudes for bound states.

Pseudospin symmetry (PSS) is a relativistic dynamical symmetry connected with the lower component of the Dirac spinor. Here, we investigate the conservation and breaking of PSS in the single-nucleon resonant states, as an example, using Green's function method that provides a novel way to precisely describe not only the resonant energies and widths but also the spacial density distributions for both narrow and wide resonances. The PSS conservation and breaking are perfectly displayed in the evolution of resonant parameters and density distributions with the potential depth: In the PSS limit, i.e., when the attractive scalar and repulsive vector potentials have the same magnitude but opposite sign, PSS is exactly conserved with strictly the same energy and width between the PS partners as well as identical density distributions of the lower components. As the potential depth increases, the PSS is broken gradually with energy and width splittings and a phase shift in the density distributions.

Pseudospin symmetry (PSS) plays critical roles in the formation of many physical phenomena, and has been studied extensively from spherical nuclei to deformed nuclei. However, there remains an open question about the PSS in resonant states in deformed systems. For the first time, we investigate the PSS in resonant states in deformed nuclei by solving the Dirac equation in the complex-momentum representation for a potential with quadrupole deformation at the first order obtained from relativistic mean-field (RMF) calculations. Taking 170Yb as an example, we explore the PSS in the single-particle resonant states by examining the energies, widths and the density distribution. Compared with the bound states, the pseudospin splittings between the resonant pseudospin doublets are found to be small, and the radial density distributions of pseudospin doublets look very similar, which indicates that the PSS of resonant states is well reserved in deformed nuclei. Moreover, we have comprehensively studied the dependencies of the PSS on nuclear deformations and the shape of potential, which are helpful to recognize the evolution of resonant pseudospin doublets in weakly-bound deformed nuclei and understand their exotic properties.

Spectrum of the doubly heavy tetraquarks, bbq¯q¯, is studied in a constituent quark model. Four-body problem is solved in a variational method where the real scaling technique is used to identify resonant states above the fall-apart decay thresholds. In addition to the two bound states that were reported in the previous study we have found several narrow resonant states above the BB⁎ and B⁎B⁎ thresholds. Their structures are studied and are interpreted by the quark dynamics. A narrow resonance with spin-parity JP=1+ is found to be a mixed state of a compact tetraquark and a B⁎B⁎ scattering state. This is driven by a strong color Coulombic attraction between the bb quarks. Negative-parity excited resonances with JP=0−, 1− and 2− form a triplet under the heavy-quark spin symmetry. It turns out that they share a similar structure to the λ-mode of a singly heavy baryon as a result of the strongly attractive correlation for the doubly heavy diquark.

The complex momentum representation method is used to explore the single particle resonances in nuclei. The energies, widths, and wavefunctions are obtained by solving the Dirac equation with a Woods-Saxon potential, in which the parameters are determined from the RMF calculations for 132Sn. The pseudospin symmetry is investigated for the resonant states in comparison with bound states. Approximate degeneracy is shown in the energies, widths, and wavefunctions between the pseudospin doublets including the resonant and bound states. The influence of the potential shape on the pseudospin symmetry is checked. The quality of pseudospin symmetry is shown to be dependent of the parameters, which is explained in terms of the change of potential with these parameters. The energy and width splittings between the pseudospin doublets are sensitive to the potential parameters, which is clarified by the change of the derivative of potential with r. Moreover, the pseudospin splitting in the wavefunctions and its dependence on potential shape are analyzed. The wave function splitting between the pseudospin doublets is quantitatively obtained in the momentum space for the resonant states because they are local and square integrable. Considering that a real nuclear field is similar to a Woods-Saxon potential, the change of the parameters in Woods-Saxon potential can reflect the change of nuclear field from stable nuclei to drop-line nuclei. Therefore, the present researches are helpful to understand the level structure of the weakly bound nuclei and their exotic properties.

We present a model of impurity-induced magnetization of graphene assuming that the main source of graphene magnetization is related to impurity states with a localized spin. The analysis of solutions of the Schrödinger equation for electrons near the Dirac point has been performed using the model of massless fermions. For a single impurity, the solution of Schrödinger’s equation is a linear combination of Bessel functions. We found resonance energy levels of the non-magnetic impurity. The magnetic moment of impurity with a localized spin was accounted for the calculation of graphene magnetization using the Green’s function formalism. The spatial distribution of induced magnetization for a single impurity is obtained. The energy of resonance states was also calculated as a function of interaction. This energy is depending on the impurity potential and the coupling constant of interaction.

Resonant states (RS), also known as quasi-normal modes, arise in spectral expansions of linear response functions of open systems. Manipulation of these spatially ‘divergent’ oscillating functions requires a departure from the usual definitions of inner product, normalization and orthogonality typical in the studies of closed systems. A multipolar Gaussian regularization method for RS inner products is introduced in the context of light scattering and shown to provide analytical results for the crucial RS inner product integrals in the problematic region exterior to the scattering system. We detail the applicability of this method to arbitrary scattering geometries while providing semi-analytic benchmark results for spherical scatterers. This formulation is then used to highlight the lack of ‘convergence’ in directly truncated RS spectral expansions and the necessity of adding non-resonant contributions to the RS spectral expansions. Solutions to these difficulties are illustrated in the case of dispersive media spheres, but these methods should prove generalizable to arbitrary RS spectral expansions.