On equivalent methods for functional determinants

• URL: http://arxiv.org/abs/2601.08686v1
• Date: Tue, 13 Jan 2026 16:11:38 GMT
• Title: On equivalent methods for functional determinants
• Abstract summary: We show how the Gel'fand-Yaglom theorem and the Green's function method can be used for computing ratios of determinants of one-dimensional operators.<n>We also comment on the presence of vanishing as well as negative eigenvalues and show how the Green's function method provides a natural prescription for handling them.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Computing functional determinants of differential operators is central to any field-theoretical calculation relying on a saddle-point expansion. A variety of approaches is available for the computation that avoid having to know the eigenspectrum of the operator, and in particular the Gel'fand-Yaglom theorem and the Green's function method. In this note, we show how both approaches can be constructed using a contour integral argument and conclude that these are completely equivalent for computing ratios of determinants of one-dimensional operators. Furthermore, we comment on the presence of vanishing as well as negative eigenvalues and show how the Green's function method provides a natural prescription for handling them.

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