The spectral determinant for second-order elliptic operators on the real line

We derive an expression for the spectral determinant of a second-order elliptic differential operator  \( \mathcal {T} \) $\mathcal{T}$ defined on the whole real line, in terms of the Wronskians of two particular solutions of the equation \( \mathcal {T} u=0\) $\mathcal{T}u=0$ . Examples of application of the resulting formula include the explicit calculation of the determinant of harmonic and anharmonic oscillators with an added bounded potential with compact support.