The Fredholm Determinant for a Dirac Operator

Abstract

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 σ 1 + mσ 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, σ i , constitute a representation of the Dirac matrices. With H 0 = p x σ 1 + mσ 3 and z a complex number, the Fredholm determinant is denoted by Det[( z− H)/( z− H 0)]. Let M( x) be the 2 × 2 matrix that transfers a spinor solution, ψ( x), of the Dirac equation Hψ( x) = zψ( x) from − L to x:ψ( x) = M( x)ψ(− L) and let M 0( x) be the corresponding matrix for H 0. Then it is shown, for eigenfunctions obeying the periodic boundary condition ψ( L) = ψ(− L), that Det[( z− H)/( z− H 0] equals the determinant of the 2 × 2 matrix [1− M( L)]/[1− M 0( L)]. The calculation of an infinite determinant is thus reduced to the calculation of a 2 × 2 determinant and for piecewise constant potentials an expression for 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, Ann. Phys. 226 (1993), 271.