The Spectrum Zero Problem of nonlinear Dirac equation with particle-antiparticle interaction

In this study, we investigate the Spectrum Zero Problem of nonlinear Dirac equations with a focus on the behavior of zero at the boundaries of the spectral gap. We introduce a nonlinear particle-antiparticle interaction and demonstrate that the problem exhibits asymmetric behavior at the left and right boundaries of the spectrum. Specifically, when zero is at the right boundary, the problem has only trivial solutions and is identified as a bifurcation point on the left, whereas nontrivial solutions exist when zero is at the left boundary or within the spectral gap. The main idea is to employ a variational method involving a perturbation technique that places zero within the spectral gap. We use the critical point theorem of the perturbed functional to construct a Palais-Smale sequence in order to approach the critical point of the target energy functional. Additionally, we utilize the concentration-compactness principle to identify critical points of the original functional and explore the associated bifurcation phenomena. Our results reveal an asymmetric phenomenon in nonlinear quantum systems and provide insights into why strongly indefinite problems typically address zero only at the left boundary of the spectral gap.