Spectral Theory in Hilbert Space

68 10. BOUNDARY CONDITIONS
Hint: If −u ′′ + qu = µu, write this as −u ′′ + (q − λ)u = (µ − λ)u
and use the variation of constants formula, thinking of (µ − λ)u as an
inhomogeneous term, to write down an integral equation for u in terms
of solutions of −v ′′ + qv = λv. Using an initial point sufficiently close
to an endpoint use estimates in this integral equation to show that u
is square integrable near the endpoint.
Exercise 10.3. Show that
[u1, v1][u2, v2] − [u1, v2][u2, v1] = [u1, u2][v1, v2]
for differentiable functions u1, u2, v1, v2. Next show that if [v1, v2] = 1,
then the boundary form for u1, u2 ∈ D1 at b equals limb([u1, v1][u2, v2]−
[u1, v2][u2, v1]). Furthermore, show that if −v ′′ + qv = λv and −u ′′ +
qu = f then ([u, v]) ′ = (f − λu)v. Finally show that if all solutions of
−v ′′ + qv = λv are in L 2 (a, b) and if u ∈ D1 then the limit at b of [u, v]
exists.
Conclude that in the case n+ = n− = 2 selfadjoint boundary conditions
may be described by conditions on the values of [u, v1], [u, v2] at
the endpoints of exactly the same form as we described them for the
regular case on the values of u, u ′ .
Exercise 10.4. Show that (10.3) is in the limit point condition at
∞ if q = q0 + q1 where q0 is bounded from below and q1 ∈ L 1 (0, ∞).
Hint: First show that |u(x)| 2 ≤ 2 x
0 |u′ ||u| if u(0) = 0, then multiply
by |q1| and integrate. Conclude that there is a constant A so that
x
0 |q1||u| 2 ≤ x
0 |u′ | 2 + A x
0 |u|2 for all x > 0. Now show, similar to the
proof of Theorem 10.10, that |u| 2 is increasing if u(0) = 0, u ′ (0) = 1
and u satisfies −u ′′ + qu = λu for an appropriate λ.