Closed-form reduction of the direct contact contribution

Determine whether the finite-interval one-dimensional time integral representing the direct contact contribution to the bound-state survival amplitude of the one-dimensional attractive delta-function well in a uniform dc electric field admits a further closed-form reduction, in particular to a single expression involving the Faddeeva function.

Background

The paper derives an exact one-dimensional time-integral representation for the direct contact contribution to the physical bound-state survival amplitude. The spatial integrations have already been evaluated analytically, leaving a finite-time integral involving products of Gaussian–complementary-error-function blocks, with simplified parameter dependence for a nonzero dc field.

The authors explicitly raise the unresolved question of whether this remaining integral can be reduced further to a single standard Faddeeva-function expression. They state that no such reduction is apparent for a general nonzero dc field and retain the one-dimensional integral as the final exact representation. This is a concrete unresolved analytical question rather than a general aspiration, because the paper supplies the exact integral whose further closed-form evaluation is unknown.