The problem of complete separation of variables in the squarable Dirac equation (Q1896711)

The subject of our consideration is the Dirac equation for an external electromagnetic field in the curved space-time \(V_4\): \[ \begin{aligned} (\widehat{H} - m)\widehat{\psi} &= (\widehat{\gamma}^i(x) \widehat{P}_i - m) \widehat{\psi} = 0,\\ [\widehat{\gamma}^i \widehat{\gamma}^j] = 2 \widehat{E}_4g^{ij}(x),\quad \widehat{P}_j &= i(\nabla_j - \widehat{\Gamma}_j) - A_j,\quad \widehat{\Gamma}_j = \widehat{\gamma}^l \widehat{\gamma}^k \nabla_j e_{a|\ell} e^{a|}_k,\end{aligned}\tag{1} \] where \(i,j,k,l = 1,2,3,4\), \(g_{ij}\), \(g^{ij}\) are components of the metric tensor, \(A_i\) is a vector potential of an electromagnetic field, \(\widehat{E}_4\) is the identity matrix, \(e^i_{a|}\) is an orthogonal tetrad, \(\nabla_i\) is the covariant derivative; when the same index appears as a subscript and superscript in a term, this term stands for the sum of the terms obtained by giving the index each of its four values. Many works are known to be devoted to the problem of complete separation of variables in equation (1). All these works are conceptually based on the following definition. Definition 1. Equation (1) admits a complete separation of variables if there exists a privileged coordinate system \(\{u^i\}\) such that the fundamental matrix of (1) referred to \(\{u^i\}\) has the form \[ \widehat {\varphi} = \widehat{S}(u) \prod^4_{i = 1} \widehat{\varphi}_i(u^i, \lambda_j),\tag{2} \] where \(\lambda_j = \) const, \([\widehat{\varphi}_i \widehat{\varphi}_j]= 0\), \(\text{det}|\widehat{S} |\neq 0\), \(\text{det}\left|{\partial \over \partial\lambda_i} \left({\partial \widehat{\Omega}\over \partial u^j} \widehat{\Omega}^{-1}\right)\right|\neq 0\), where \(\widehat \Omega = \prod^4_{i = 1} \widehat{\varphi}_i(u^i, \lambda_j)\) (except where the problem of complete separation of variables in the diagonalized squarable Dirac equation is considered). In the present article we sugest the following generalization of Definition 1. Definition 2. Equation (1) admits a complete separation of variables if there exists a privileged coordinate system \(\{u^i\}\) such that the fundamental matrix has the form \[ \widehat{\psi} = (\widehat {H} + m) \widehat{S} \widehat{\Omega}, \] where the matrix \(\widehat{\Omega}\) is defined in (2).