Spectral Theory in Hilbert Space

13. FIRST ORDER SYSTEMS 95
and (u2, v2) ∈ T1. Then the boundary form (cf. Chapter 9) is
(13.3) 〈(u1, v1), U(u2, v2)〉 = i
= i
I
I
(v ∗ 2W u1 − u ∗ 2W v1)
((Ju ′ 2 + Qu2) ∗ u1 − u ∗ 2(Ju ′ 1 + Qu1))
= −i
I
(u ∗ 2Ju1) ′ = −i lim
K→I [u ∗ 2Ju1]K,
the limit being taken over compact subintervals K of I. We must
restrict T1 so that this vanishes. Like in Chapter 10 this means that
the restriction of T1 to a selfadjoint relation T is obtained by boundary
conditions since the limit clearly only depends on the values of u1 and
u2 in arbitrarily small neighborhoods of the endpoints of I.
An endpoint is called regular if it is a finite number and Q and W
are integrable near the endpoint. Otherwise the endpoint is singular.
If both endpoints are regular, we again say that we are dealing with
a regular problem. We have a singular problem if at least one of the
endpoints is infinite, or if at least one of Q and W is not integrable on
I.
Consider now the regular case. Since it is clear that both deficiency
indices equal n in the regular case there are always selfadjoint
realizations. To see what they look like, let ũ be the boundary value
of (u, v) ∈ T1, i.e., ũ =
so that the
u(a)
u(b)
. Also put B = iJ 0
0 −iJ
boundary form is ũ ∗ 2Bũ1. Now if u ∈ Di then 〈u, Uu〉 = 〈u, u〉 so that
the boundary form is positive definite on Di. Similarly it is negative
definite on D−i (cf., Corollary 9.17). Since dim Di ⊕D−i = 2n the rank
of the boundary form is 2n on this space so that the boundary values
of this space, and a fortiori those of T1, range through all of C 2n . Since
〈T1, UT0〉 = 0 it follows that the boundary value of any element of T0
is 0.
Conversely, to guarantee that 〈T1, Uu〉 = 0 for some u ∈ T1 it is
obviously enough that the boundary value of u vanishes. Hence the
minimal relation consists exactly of those elements of the maximal relation
which have boundary value 0. It is now clear that any maximal
symmetric restriction of T1 is obtained by restricting the boundary
values to a maximal subspace of C 2n for which the boundary form vanishes,
a so called maximal isotropic space for B. We know, since the
deficiency indices are finite and equal, that all such maximal symmetric
restrictions are actually selfadjoint (Corollary 9.15). Since the problem
of finding maximal isotropic spaces of B is a purely algebraic one
we consider the problem of identifying all selfadjoint restrictions of T1
solved in the regular case. See also Exercise 13.4.