Perturbation method for a class of nonlinear Dirac equations. (English) Zbl 0974.35101

The author studies the solution \(\psi(x_0,x)=\exp(i\omega x_0)\Phi(x)\) of the Dirac equations \[ i\sum^3_{j= 0}\gamma^j\partial_{x_j}\psi- m\psi+ \gamma^0\nabla F(\psi)= 0,\quad F(\psi)= \{G(\overline\psi\psi)+ H(\overline\psi \gamma^5\psi)\}/2 \] under the conditions \(G(0)= H(0)= 0\), \(g(at)= G'(at)= a^\theta g(t)\), \(h(at)= H'(at)= a^\theta h(t)\), \(0\leq\theta< 1\). When \(\Phi(x)= (\alpha\phi(\lambda x),\beta\chi(\lambda x))\); \(\alpha= \lambda\beta/(m- \omega)\) etc., the system \[ (\sigma p)\chi- \phi+ g(\phi^2)\phi+ K_1(\varepsilon, \phi,\chi)= 0,\quad (\sigma p)\phi+ 2m\chi+ K_2(\varepsilon, \phi,\chi)= 0 \] expressed by \(D(\varepsilon, \phi,\chi)= 0\) is derived. Here \(\sigma p= i\sum_{j=1}^3\sigma^j\partial_{x_j}\), \(\varepsilon= m-\omega\geq 0\), and \(D:\mathbb{R}\times X\times X\to Y\times Y\), where \(X= H^1(\mathbb{R}^3, C^2)\), \(Y= L^2(\mathbb{R}^3, C^2)\). When \(\varepsilon= 0\), the system becomes \(-\Delta\phi/(2m)+ \phi- g(\phi^2)\phi= 0\), \(\chi= -(\sigma p)\phi/(2m)\) with the solution \((\phi_0,\chi_0)\). Let \(X_r\subset X\times X\), \(N= \text{Ker }D_{\phi,\chi}(0, \phi_0,\chi_0)\) and \(N\oplus N^\perp= X\times X\).
Result 1. When \(h\equiv 0\), there exist \(\delta> 0\) and \(\eta\in C((0,\delta),X_r)\) such that \(\eta(0)= (\phi_0,\chi_0)\) and \(D(\varepsilon, \eta(\varepsilon))= 0\).
Result 2. There exist \(\delta>0\), a nbd. \(W_0\) of \(0\) in \(N\) and \(\eta\in C^1((0,\delta)\times W_0, N^\perp)\) such that \(\eta(0,0)= (\phi_0, \chi_0)\) and \(D(\varepsilon, Q_1+\eta(\varepsilon, Q_1))= 0\) for \(Q_1\in W_0\).


35Q40 PDEs in connection with quantum mechanics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35B20 Perturbations in context of PDEs