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 $\mathcal{T}$ defined on the whole real line, in terms of the Wronskians of two particular solutions of the equation $\mathcal{T}u=0$. 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.