= Instanton fluctuation prefactor
{title2=$K=\sqrt{S_I/(2\pi\hbar)}\,[\det L_0/\det' L_I]^{1/2}$}
For a one-dimensional <quantum tunnelling> <instanton>, the prefactor combines nonzero Gaussian fluctuation <eigenvalues> with the translation zero-mode Jacobian. Here $L_I=-\partial_t^2+V''(q_I)/m$, $L_0=-\partial_t^2+\omega^2$, and the prime excludes the zero mode. Common endpoint regularization and reference normalization define the <determinant> ratio. The resulting $K$ has frequency units, while $Ke^{-S_I/\hbar}$ is the dilute crossing rate. This <determinant>/collective-coordinate structure is also discussed in https://arxiv.org/pdf/2001.10128[Dunne, Sulejmanpasic and Ünsal's treatment]; unequal neighbouring well frequencies require care and need not share the symmetric-double-well result.
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