This monograph develops the spectral theory of an
$n$th order non-self-adjoint two-point differential operator
$L$ in the Hilbert space $L^2[0,1]$. The
mathematical foundation is laid in the first part, where the spectral
theory is developed for closed linear operators and Fredholm
operators. An important completeness theorem is established for the
Hilbert-Schmidt discrete operators. The operational calculus plays a
major role in this general theory.
In the second part, the spectral theory of the differential
operator $L$ is developed by expressing $L$ in the
form $L = T + S$, where $T$ is the principal part
determined by the $n$th order derivative and $S$ is
the part determined by the lower-order derivatives. The spectral
theory of $T$ is developed first using operator theory, and
then the spectral theory of $L$ is developed by treating
$L$ as a perturbation of $T$. Regular and irregular
boundary values are allowed for $T$, and regular boundary
values are considered for $L$. Special features of the
spectral theory for $L$ and $T$ include the
following: calculation of the eigenvalues, algebraic multiplicities
and ascents; calculation of the associated family of projections which
project onto the generalized eigenspaces; completeness of the
generalized eigenfunctions; uniform bounds on the family of all finite
sums of the associated projections; and expansions of functions in
series of generalized eigenfunctions of $L$ and $T$.
Readership
Graduate students and research mathematicians interested in
ordinary differential equations.