On the nature of near-threshold bound and virtual states

Physical states are characterised uniquely by their pole positions and the corresponding residues. Accordingly, in those parameters also the nature of the states should be encoded. For bound states (poles on the real s-axis below the lowest threshold on the physical sheet) there is an established criterion formulated originally by Weinberg in the 1960s, which allows one to estimate the amount of compact and molecular components in a given state. We demonstrate in this paper that this criterion can be straightforwardly extended to shallow virtual states (poles on the real s-axis below the lowest threshold on the unphysical sheet) which should be classified as molecular. We argue that predominantly non-molecular or compact states exist either as bound states or as resonances (poles on the unphysical sheet off the real energy axis) but not as virtual states. We also discuss the limitations of the mentioned classification scheme.

Vacuum polarization of planar charged fermions with Coulomb and Aharonov–Bohm potentials

Vacuum polarization of charged massless fermions is investigated in the superposition of Coulomb and Aharonov–Bohm (AB) potentials in 2 + 1 dimensions. For this purpose, we construct the Green function of the two-dimensional Dirac equation with Coulomb and AB potentials (via the regular and irregular solutions of the radial Dirac equation) and then calculate the vacuum polarization charge density in the so-called subcritical and supercritical regimes. In the supercritical regime, the Green function has a discontinuity in the complex plane of “energy” due to the singularities on the negative energy axis; these singularities are situated on the unphysical sheet and related to the creation of infinitely many quasistationary fermionic states with negative energies. We expect that our results will be helpful in gaining deeper understanding of the fundamental problem of quantum electrodynamics which can be applied to the problems of charged impurity screening in graphene taking into consideration the electron spin.

$\bar KNN$ RESONANCE $\bar KNN - \pi YN$ COUPLED CHANNEL FADDEEV EQUATION

The three-body resonance of [Formula: see text] system is investigated by using the [Formula: see text] coupled channels Faddeev equation. The resonance energy is determined from the pole of S -matrix on the unphysical sheet. It is found that the pole positions of the predicted amplitudes are significantly modified when the three-body dynamics is approximately treated by introducing the effective [Formula: see text] two-body interaction.

Resonances of a λ-rational Sturm–Liouville problem

We consider a family of self-adjoint 2 × 2-block operator matrices Ãϑ in the space L2(0, 1) ⊕ L2(0, 1), depending on the real parameter ϑ. If Ã0 has an eigenvalue that is embedded in the essential spectrum, then it is shown that for ϑ ≠ 0 this eigenvalue in general disappears, but the resolvent of Ãϑ has a pole on the unphysical sheet of the Riemann surface. Such a pole is called a resonance pole. The unphysical sheet arises from analytic continuation from the upper half-plane ℂ+ across the essential spectrum. Furthermore, the asymptotic behaviour of this resonance pole for small ϑ is investigated. The results are proved by considering a certain λ-rational Sturm-Liouville problem and its Titchmarsh–Weyl coefficient.

Resonances of a λ-rational Sturm–Liouville problem

We consider a family of self-adjoint 2 × 2-block operator matrices Ãϑ in the space L2(0, 1) ⊕ L2(0, 1), depending on the real parameter ϑ. If Ã0 has an eigenvalue that is embedded in the essential spectrum, then it is shown that for ϑ ≠ 0 this eigenvalue in general disappears, but the resolvent of Ãϑ has a pole on the unphysical sheet of the Riemann surface. Such a pole is called a resonance pole. The unphysical sheet arises from analytic continuation from the upper half-plane C+ across the essential spectrum. Furthermore, the asymptotic behaviour of this resonance pole for small ϑ is investigated. The results are proved by considering a certain λ-rational Sturm–Liouville problem and its Titchmarsh–Weyl coefficient.