Chapter 6 develops the spectral theory for the general differential operator L
determined by regular boundary values. The spectral theory of L mirrors that of
its principal part T. In developing this spectral theory perturbation techniques are
used. Very little is known about the case of general L subject to irregular boundary
values, but the case n 2 has been recently analyzed in the series [26—29].
The spectral theory of two-point differential operators was begun by Birkhoff
in his two papers [3, 4] of 1908, where he introduced regular boundary values for
the first time. It was continued by Stone [38, 39] with the initial work on irregular
boundary values, and by Hoffman [12] in his thesis which examined second order
differential operators under irregular boundary values. Much of the spectral theory
for regular boundary values is also given in Naimark [31]. In Chapter XIX of their
treatise [6], Dunford and Schwartz give a modern operator theoretic development
of the spectral theory for regular boundary values; it includes the
functions in terms of the generalized eigenfunctions. Benzinger [2] and Schultze [36]
have studied Riesz summability of eigenfunction expansions in the case of special
classes of irregular boundary values. These references are but a few in the extensive
literature on the spectral theory of differential operators. They represent the work
that the author is most familiar with, and that has most directly influenced his own
research. Each of these references contains a bibliography which can be used as a
guide to the literature (see especially [6, pp. 2371-2374]). The author's research
in this area is contained in the references [18—30]; much of it is coauthored with
Patrick Lang.
Let us briefly discuss the relationship between Chapter XIX of Dunford and
Schwartz [6] and this monograph. First, their treatment of the spectral theory of
differential operators is based on the theory of unbounded spectral operators, which
they earlier develop in Chapters XV-XVIII. They consider only regular boundary
values. Our approach is based on the theory of Fredholm operators and on the
characteristic determinant and the Green's function; it uses only basic operator
theory. We consider not only regular boundary values, but also include the irregular
boundary values wherever possible.
Second, the multiple eigenvalue case is introduced in [6, p. 2324] as Case l.B
where (3 = ±1, but it is never mentioned again. An explanation for this is given
in [25] for the case of the formal differential operator (d/dt)2 subject to regular
boundary conditions. For a special class of regular boundary values, it is shown
that the associated projections are unbounded, and hence, the theory of spectral
operators can not be used in their study. However, by using a pairwise grouping of
the projections, we are able to produce a family of uniformly bounded projections,
and these differential operators do have a complete spectral theory which closely
resembles that of spectral operators. These ideas are generalized here to include
the nth order case (see Case 2 that appears in §7 of Chapter 4 and in §§1 and 3 of
Chapter 5).
Third, the completeness theory appearing on pp. 2334 and 2341 of [6] is incom-
plete because the basis functions 7/c(£, /i), k = 0 , 1 , . . . , n 1, are not bounded on
the sectors A and Ai, A2. We correct this problem by altering the bases and sectors
to produce the boundedness needed to estimate the Green's function and resolvent.
In establishing completeness of the generalized eigenfunctions, we use Theorem 6.2
of Chapter 2. This result is Corollary XI.6.31 of [6], which surprisingly is never
used by them in their treatment of completeness.
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