Social copying drives a tipping point for nonlinear population collapse

Abstract

We study spectral properties of Dirac operators on bounded domains Ω ⊂ R3 with boundary conditions of electrostatic and Lorentz scalar type and which depend on a parameter τ∈ R; the case τ= 0 corresponds to the MIT bag model. We show that the eigenvalues are parametrized as increasing functions of τ, and we exploit this monotonicity to study the limits as τ→ ± ∞. We prove that if Ω is not a ball then the first positive eigenvalue is greater than the one of a ball with the same volume for all τ large enough. Moreover, we show that the first positive eigenvalue converges to the mass of the particle as τ↓ - ∞, and we also analyze its first order asymptotics. © 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.