The Fredholm determinant for a Dirac operator (Q1323818)

The Fredholm determinant for a Dirac operator appropriate to a particle moving in one spatial dimension is investigated. The operator is written as \(H=p_ x\sigma_ 1+ m\sigma_ 3+ V(x)\), where \(p_ x\), \(m\) and \(V(x)\) are, respectively, the momentum, mass, and potential energy of the particle and the Pauli spin matrices, \(\sigma_ i\), constitute a representation of the Dirac matrices. With \(H_ 0= p_ x \sigma_ 1+ m\sigma_ 3\) and \(z\) a complex number, the Fredholm determinant is denoted by \(\text{Det} [(z- H)/(z- H_ 0)]\). Let \(M(x)\) be the \(2\times 2\) matrix that transfers a spinor solution, \(\psi(x)\), of the Dirac equation \(H\psi(x)= z\psi(x)\) from \(-L\) to \(x: \psi(x)= M(x)\psi(-L)\) and let \(M_ 0(x)\) be the corresponding matrix for \(H_ 0\). Then it is shown, for eigenfunctions obeying the periodic boundary condition \(\psi(L)= \psi(-L)\), that \(\text{Det} [(z-H)/ (z-H_ 0)]\) equals the determinant of the \(2\times 2\) matrix \([1- M(L)]/ [1-M_ 0(L)]\). The calculation of an infinite determinant is thus reduced to the calculation of a \(2\times 2\) determinant and for piecewise constant potentials an expression for \(\text{Det} [(z-H)/ (z-H_ 0)]\) may be derived in closed form. The relation between the Fredholm determinant and the finite determinant was conjectured in an earlier work by D. Waxman and K. D. Ivanova-Moser.