Determinants, finite-difference operators and boundary value problems

Abstract

We relate the determinants of differential and difference operators to the boundary values of solutions of the operators. Previous proofs of related results have involved considering one-parameter families of such operators, showing the desired quantities are equal up to a constant, and then calculating the constant. We take a more direct approach. For a fixed operator, we prove immediately that the two sides of our formulas are equal by using the following simple observation (Proposition 1.3):Let U∈SU(n,C).Write U in block form

$$U = \left( {\begin{array}{*{20}c} {u_{11} } & {u_{12} } \\ {u_{21} } & {u_{22} } \\ \end{array} } \right),$$

where u ₁₁ and u ₂₂ are square matrices. Then

$$\det u_{11} = \overline {\det u_{22} } .$$