Spectral stability of small amplitude solitary waves of the Dirac equation with the Soler-type nonlinearity

摘要: We study the point spectrum of linearization at a solitary wave solution $\phi_\omega(x)e^{-\mathrm{i}\omega t}$ to nonlinear Dirac equation in $\mathbb{R}^n$, $n\ge 1$, with term given by $f(\psi^*\beta\psi)\beta\psi$ (known as Soler model). focus on spectral stability, that is, absence eigenvalues nonzero real part, non-relativistic limit $\omega\lesssim m$, case when $f\in C^1(\mathbb{R}\setminus\{0\})$, $f(\tau)=|\tau|^k+O(|\tau|^K)$ for $\tau\to 0$, $0 4/n$. An important part stability analysis is proof bifurcations nonzero-real-part from embedded threshold points $\pm 2m\mathrm{i}$. Our approach based constructing new family exact bi-frequency solutions model, using this determine multiplicity 2\omega\mathrm{i}$ linearized operator, and behaviour "nonlinear eigenvalues" (characteristic roots holomorphic operator-valued functions).