Transactions of the AMS. Year 1961. Volume 101. Number 02.

Some new analytical techniques and their application to irregular cases for the third order ordinary linear boundary-value problem
Nathaniel R. Stanley
351-376

Abstract: 1. For the operator $T_3^ - (D)$ defined by $- {d^3}/d{x^3}$ and a triple of boundary conditions irregular in the sense of Birkhoff, the reduction of this triple to canonical forms is implicit in the reduction made for a more general third order operator (Theorem 1.2). 2. A new technique is developed for calculating the Green's function for the nth order ordinary linear boundary-value problem (Theorem 2.4), and is applied to $T_3^ -$; a necessary and sufficient condition is given for the identification of degenerate sets of boundary conditions for $T_3^ -$ (Theorem 2.6). 3. A new technique is developed for calculating asymptotic expansions for large zeros of exponential sums, and the form of the expansion, which includes a logarithmic asymptotic series, is established by induction (Theorem 3.1); expansions for the cube roots of the eigenvalues of $ T_3^ -$ then follow as special cases. 4. A theorem of Dunford and Schwartz (Theorem 4.0) giving a sufficient condition for completeness of eigenfunctions in terms of growth of the norm of the resolvent operator, is applied to prove that, with a possible exception, the eigenfunctions of $T_3^ -$ span $ {L_2}(0,1)$ (Theorem 4.5).