The Spectral Theory of Second Order Two-Point Differential Operators II. Asymptotic Expansions and the Characteristic Determinant

In this second paper of a four-part series, we construct the characteristic determinant of a two-point differential operator L in L2[0, 1], where L is determined by ℓ = −D2 + q and by independent boundary values B1, B2. For the solutions u(•; ρ) and υ(•;ρ) of the differential equation ρ2u + u″−qu = 0 with u(1; ρ) = eiρ, u′(1; ρ) = iρeiρ and υ(0; ρ) = 1, υ′(0; ρ) = −iρ, asymptotic formulas are established on a half plane Im ρ ≥ −d. Then u(•; ρ) and υ(•; ρ) lead to the characteristic determinant Δ(ρ) of L, which also satisfies an asymptotic formula on Im ρ ≥ −d.