(Modified) Fredholm Determinants for Operators with Matrix-Valued Semi-Separable Integral Kernels Revisited

We revisit the computation of (2-modified) Fredholm determinantsfor operators with matrix-valued semi-separable integral kernels. The latteroccur, for instance, in the form of Greens functions associated with closedordinary differential operators on arbitrary intervals on the real line. Ourapproach determines the (2-modified) Fredholm determinants in terms of solutionsof closely associated Volterra integral equations, and as a result offersa natural way to compute such determinants.We illustrate our approach by identifying classical objects such as theJost function for half-line Schrödinger operators and the inverse transmissioncoe.cient for Schrödinger operators on the real line as Fredholm determinants,and rederiving the well-known expressions for them in due course.We also apply our formalism to Floquet theory of Schrödinger operators, andupon identifying the connection between the Floquet discriminant and underlyingFredholm determinants, we derive new representations of the Floquetdiscriminant.Finally, we rederive the explicit formula for the 2-modified Fredholmdeterminant corresponding to a convolution integral operator, whose kernelis associated with a symbol given by a rational function, in a straghtforwardmanner. This determinant formula represents a Wiener-Hopf analog of Daysformula for the determinant associated with finite Toeplitz matrices generatedby the Laurent expansion of a rational function.