Resummed heat-kernel and form factors for surface contributions: Dirichlet semitransparent boundary conditions

In this article we consider resummed expressions for the heat-kernel's
trace of a Laplace operator, the latter including a potential and imposing Dirichlet semitransparent boundary conditions on a surface of codimension one in flat space.
We obtain resummed expressions that correspond to the first and second order expansion of the heat-kernel in powers of the potential.
We show how to apply these results to obtain the bulk and surface form factors of a scalar quantum field theory in $d=4$ with a Yukawa coupling to a background.
A characterization of the form factors in terms of pseudo-differential operators is given.