x PREFACE
operator L is introduced in the Hilbert space
L2[a,
6], and an overview of its spectral
theory is given. Fundamental to this discussion is the Sobolev space Hn[a,b] and
its associated Sobolev structure. Since L is a Fredholm operator of index zero, we
are able to characterize the spectrum of L using the general results of Chapter 2.
The characteristic determinant D is defined in its initial form, and the eigenvalues
of L are shown to be the zeros of D. A key result is that the algebraic multiplicity
of an eigenvalue is equal to its order as a zero of D.
The spectral theory in Chapters 4-6 is set in the Hilbert space L2[0,1], and the
differential operator L is expressed in the form
L = T + S,
where T is the principal part of L determined by the nth order derivative and S is
the part determined by the lower order derivatives. The differential operator T is
of great interest in its own right; it serves as a model for the general spectral theory
of differential operators.
The spectral theory of T is established in Chapters 4 and 5. Included are the
following topics:
(i) Asymptotic formulas for the characteristic determinant A of T and for the
Green's function G( •; •; A) of XI T. These quantities are simpler when expressed
in terms of the p variable where A =
pn.
(ii) Classification of the boundary values B\,... , Bn determining T as being
either regular, irregular, or degenerate, depending on the form of the characteristic
determinant.
(iii) Calculation of the eigenvalues of T by calculating the zeros of A. For
the eigenvalues asymptotic formulas are derived, and the corresponding algebraic
multiplicities and ascents are determined.
(iv) Calculation of the family of projections V associated with T, and formation
of the corresponding subspaces Soc(T) and Moo(T). The projections in V map
L2[0,1]
onto the generalized eigenspaces of T, and the subspace S'oo(T) consists of
all functions in 1? [0,1] that can be expressed in a series of generalized eigenfunctions
of T.
(v) Development of decay rates for the resolvent R\(T) along rays from the
origin, thereby showing that
S^(T)=L 2 [0,1] and M^T) = {0}.
This is accomplished using the completeness theorem of Chapter 2; it shows that
the generalized eigenfunctions of T are complete in L2[0,1]; and it is valid for both
regular and irregular boundary values.
(vi) Demonstrating that the family of all finite sums of the projections in V
is uniformly bounded in norm. Here it is assumed that the boundary values are
regular.
(vii) Establishing that S'00(T) is a closed subspace when the boundary values
are regular, in which case each function in
L2[0,1]
can be expanded in a series of
generalized eigenfunctions of the differential operator T.
For the special case n = 2 with irregular boundary values, it is known that the
projections in V are unbounded and S^iT} is a proper dense subspace of
L2[0,1].
See [22, 23]. The situation for nth order T is unknown; we conjecture that it is
identical to the second order case.
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