# A question about Gel'fand-Yaglom method of calculating functional determinants

I know that the Gel'fand-Yaglom method is a way to calculate determinants of 1D differential operators.

For instance, let us consider an operator $-\partial_r^2+W(r)$. Let us define $\psi_{0}(r)$ as the solution of the equation $$(-\partial_r^2+W(r))\psi=0$$ with the boundary conditions $$\psi(0)=0,\ \psi'(0)=1.$$ Then the Gel'fand-Yaglom theorem states that $$\det(-\partial_r^2+W(r))=\lim_{r\rightarrow\infty}\psi_0(r),$$ where the equality should be understood in the sense of ratio.

My question is the following. I am reading this paper, where a generalized Gel'fand-Yaglom theorem was used (the author did not claim to be using the Gel'fand-Yaglom method). The generalization for fields with additional structure (spinors, vectors, etc.) is as follows: $$\frac{\det(-\partial^2+W)}{\det(-\partial^2)}=\lim_{r\rightarrow\infty}\frac{\det\psi_W(r)}{\det\psi_0(r)},$$ where the $\det$ on the right hand side is for the ordinary determinant over the residual indices (spinorial, gauge group, etc.). I have tried to search for references on such generalization but not successful. Also, it seems that the boundary conditions the authors used are different from the original theorem stated above.