Spectral Theory in Hilbert Space

94 13. FIRST ORDER SYSTEMS
n (pointwise) linearly independent solutions of these equations the deficiency
indices can be no larger than n; in particular they are both
finite. We now make the following basic assumption.
Assumption 13.7. If K is a sufficiently large, compact subinterval
of I there is no non-trivial solution of Ju ′ +Qu = 0 with
K u∗ W u = 0.
Note that if there is a solution with u ∗ W u = 0, then W u = 0 so u
actually also solves Ju ′ + Qu = λW u for any complex λ. The assumption
automatically holds if (13.1) is equivalent to a Sturm-Liouville
equation, or more generally an equation of the types discussed in Example
13.3 and Exercise 13.3. One reason for making the assumption is
that it ensures that the deficiency indices of T0 are precisely equal to the
dimensions of the spaces of those solutions of Ju ′ + Qu = ±iW u which
have finite norm, but the assumption will be even more important in
the next chapter.
According to Corollary 9.15 there will be selfadjoint realizations of
(13.1) precisely if the deficiency indices are equal. We will in the rest
of this chapter assume that a selfadjoint extension of T0 exists. Some
simple criterions that ensure this are given in the following proposition,
but if these do not apply it can, in a concrete case, be very difficult to
determine whether there are selfadjoint realizations or not.
Proposition 13.8. The minimal relation T0 has equal deficiency
indices if either of the following conditions is satisfied:
(1) J, Q and W are real-valued.
(2) The interval I is compact.
Proof. If u ∈ L 2 W satisfies Ju′ + Qu = λW u and the coefficients
are real-valued, then conjugation shows that u is still in L 2 W and Ju′ +
Qu = λW u. There is therefore a one-to-one correspondence between
Dλ and D λ which obviously preserves linear independence. It follows
that n+ = n−.
If I is compact, then solutions of Ju ′ + Qu = λW u are absolutely
continuous in I, and W is integrable in I, so that all solutions are in
L 2 W . Thus n+ = n− = n.
Example 13.9. Note that J ∗ = −J, so J can be real-valued only if
n is even (show this!). Suppose u solves the equation m
k=0 (pku (k) ) (k) =
iwu where the coefficients p0, . . . pm are realvalued and w > 0. Then u
satisfies m
k=0 (pku (k) ) (k) = −iwu. It follows that if (13.1) is equivalent
to an equation of this form, then its deficiency indices are always equal
so that selfadjoint realizations exist. This is in particular the case for
the Sturm-Liouville equation (10.1).
We will now take a closer look at how selfadjoint realizations are
determined as restrictions of the maximal relation. Suppose (u1, v1)