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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.

In case somebody is familiar with this, please help me. I would appreciate your help very much.

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