Eigenvalues and Completeness for Regular and Simply Irregular Two-Point Differential Operators

In this monograph the author develops the spectral theory for an $n$th order two-point differential operator $L$ in the Hilbert space $L^2[0,1]$, where $L$ is determined by an $n$th order formal differential operator $\ell$ having variable coefficients and by $n$ linearly independent boundary values $B_1, \ldots, B_n$. Using the Birkhoff approximate solutions of the differential equation $(\rho^n I - \ell)u = 0$, the differential operator $L$ is classified as belonging to one of three possible classes: regular, simply irregular, or degenerate irregular. For the regular and simply irregular classes, the author develops asymptotic expansions of solutions of the differential equation $(\rho^n I - \ell)u = 0$, constructs the characteristic determinant and Green's function, characterizes the eigenvalues and the corresponding algebraic multiplicities and ascents, and shows that the generalized eigenfunctions of $L$ are complete in $L^2[0,1]$. He also gives examples of degenerate irregular differential operators illustrating some of the unusual features of this class.