Creation of charged fermion pair from a vacuum in the so-called supercritical Coulomb potential is examined for the case when created pair moves in one plane. In which case the quantum dynamics of charged massive or massless fermions can be described by the two-dimensional Dirac Hamiltonians with a Coulomb potential. These Hamiltonians are singular and require the additional definition in order for them to be treated as self-adjoint quantum-mechanical operators. We construct the self-adjoint two-dimensional Dirac Hamiltonians with a Coulomb potential and determine the quantum-mechanical states for such Hamiltonians in the corresponding Hilbert spaces of square-integrable functions. We determine the scattering amplitude in which the self-adjoint extension parameter is incorporated and then obtain the equations implicitly defining the possible discrete energy spectra of the self-adjoint Dirac Hamiltonians with a Coulomb potential. It is shown that the quantum system in the presence of a supercritical Coulomb potential becomes unstable which manifests in the appearance of quasi-stationary states in the lower (negative) energy continuum. The energy spectrum of these states is quasi-discrete, consists of broadened levels whose width is related to the inverse lifetime of the quasi-stationary state as well as the creation probability of charged fermion pair by supercritical Coulomb field. Explicit analytical expressions for the creation probabilities of charged (massive or massless) fermion pair are obtained in a supercritical Coulomb field.
Supercritical Coulomb center and excitonic instability in graphene
众所周知，石墨烯中的超临界库仑杂质存在具有复能量的共振态。我们发现，准粒子间隙的开放减少了虚部的能量，|ImE|，这些国家和稳定的系统。对于石墨烯中具有强库仑相互作用的无带隙准粒子，我们求解了电子-空穴束缚态的Bethe-Salpeter方程，并证明了当耦合足够强时，该方程具有快子解，α=e^2/κ \hbar v_F导致系统不稳定.在无规相位近似下，临界耦合系数α c =1.62，与库仑中心问题中的临界电荷相似。我们认为，激子不稳定性应解决通过形成一个激子凝聚和准粒子谱中的间隙产生。
Estimating the realistic start-up of spontaneous emission in the supercritical QED
本文用非微扰方法研究了由电荷为$Z$、尺寸为$R$的超临界库仑源引起的自发正电子辐射，重点讨论了真空能量$\mathcal{E}_{VP}$ \footnote{Throughout the paper the abbreviation "VP"\, stands for "vacuum polarization"\,.}，它被认为是$R$的函数，而$Z>Z_{cr,1}\simeq 170$ $Z>Z_{cr,1}\simeq 170$是固定的.结果表明，在超临界区的行为$\mathcal{E}_{VP}(R)$原来是相当不同的库仑电荷$Z$的值的依赖。特别地，自发发射变成仅从$Z \sim 300$开始的可见效果。随着$Z$的进一步生长，其强度非常迅速地增加，这为明确的检测开辟了可能性。对于这样的$Z$，出现的自发发射图像被证明是公认的，并且就共振衰减与真空壳层形成相结合而言是相当透明的。亲密的，但至关重要的作用，$\mathcal{E}_{VP}(R)$在系统中的总能量平衡的概述。本文还讨论了由轻子数守恒引起的自发发射的附加问题。
