A new method for obtaining solutions of the Dirac equation (Q5926108)

The author is interested in the construction of explicit solutions for Dirac equations with a different kind of potentials and for stationary Schrödinger equations with potential. Equations under consideration are e.g. NEWLINE\[NEWLINE\left(\gamma_0 \partial_t- \sum^3_{k=1} \gamma_k \partial_k+ im+g(x) \right) \Phi(t,x)=0, \quad\bigl(\Delta +v(x)\bigr) u(x)=0,NEWLINE\]NEWLINE where \(\gamma_k\) are Pauli matrices, \(g\) and \(v\) are special potentials. For the Dirac equations the author wants to find solutions in the form \(\Phi(t,x) =q(x)e^{iwt}\). The method for construction of such solutions works for potentials \(g=g(\xi(x))\) and \(v=v(\xi (x))\), where \(\xi\) solves \(\Delta\xi -|\text{grad} \xi |\cdot |\text{grad} \xi |'=0\). The main tool of the approach is to use special biquaternionic projection operators, which allow to reduce Dirac equations to \((D-f_0(x)) u(x)=0\), where \(D\) is the Moisil-Theodorescu operator and \(f_0\) is a scalar potential (the solutions of \(Du=0\) are the ``holomorphic functions'' in \(\mathbb{R}^3)\).