Spectral Theory in Hilbert Space

98 13. FIRST ORDER SYSTEMS
Proof. Let [c, d] be a compact subinterval of I = (a, b) such that
d
c F ∗ (·, λ)W F (·, λ) is invertible and put v1 = v in (a, c] and v1 ≡ 0 in
[d, b). Now let
u1(x) = F (x, λ)(u(c) + J −1
x
F ∗ (y, λ)W (y)v1(y) dy) .
It is clear that u1 = u in (a, c] and if we choose v1 appropriately in
[c, d] we can achieve that u1 ≡ 0 in [d, b). In fact, setting v(x) =
−F (x, λ)( d
c F ∗ (·, λ)W F (·, λ)) −1 Ju(c) in this interval will do.
It follows that (u − u1, v − v1) ∈ T1 is 0 near a and equals (u, v)
near b. We can therefore find a maximal non-degenerate space for Ba
consisting of elements of T1 vanishing near b. Similarly, a maximal nondegenerate
space for Bb consisting of elements of T1 vanishing near a.
Thus Ba and Bb are independent, as claimed. Since the signature of
the complete boundary form Bb − Ba is (n+, n−) the independence of
Ba and Bb implies that n+ = r a − + r b + and n− = r a + + r b −, using the
notation introduced above for the signatures of Ba and Bb. According
to Corollary 9.15 T1 has selfadjoint restrictions precisely if n+ = n−.
Reasoning like in the regular case it follows that there are selfadjoint
restrictions defined by separated boundary conditions precisely if r a + =
r a − and r b + = r b −, from which n+ = n− follows. In fact, from any two of
these relations the third clearly follows.
Consider finally the case when a is a regular endpoint but b possibly
is singular. In this case Ba is given by Ba(u1, u2) = iu2(a) ∗ Ju1(a), with
notation as above. Clearly r a + is the number of positive eigenvalues of
iJ and r a − the number of negative eigenvalues. It follows that selfadjoint
restrictions of T1 defined by separated boundary conditions exist if and
only if the deficiency indices are equal and iJ has an equal number
of positive and negative eigenvalues; in particular n must be even. In
the Sturm-Liouville case all these conditions are fulfilled, as we already
know.
Exercises for Chapter 13
Exercise 13.1. Show that u and ũ are elements of the same equivalence
class in L2 W if and only if W u = W ũ a.e.
Exercise 13.2. Verify that T0 is the graph of an operator if (13.1)
is equivalent to an equation of the type (10.1) (or more generally an
equation of the type discussed in Exercise 13.3) and w > 0 a.e. in
I. Also show that in this case Assumption 13.7 holds. Try to show
this assuming only that w ≥ 0 but w > 0 on a subset of I of positive
measure (this is considerably harder).
c