The spectral theory of second order two-point differential operators: I. A priori estimates for the eigenvalues and completeness

Synopsis This paper is the first part in a four-part series which develops the spectral theory for a two-point differential operator L in L 2 [0, 1] determined by a second order formal differential operator l = −D 2 + pD + q and by independent boundary values B 1 , B 2 . The differential operator L is classified as belonging to one of five cases, Cases 1–5, according to conditions satisfied by the coefficients of B 1 , B 2 . For Cases 1–4 it is shown that if λ = ρ 2 is any eigenvalue of L with ∣ρ∣ sufficiently large, then ρ lies in the interior of a horizontal strip (Cases 1–3) or the interior of a logarithmic strip (Case 4), and in each of these cases the generalised eigenfunctions of L are complete in L 2 [0, 1].