Spectral Theory in Hilbert Space

1. ASYMPTOTICS OF THE m-FUNCTION 85
as λ → ∞ along any non-real ray 1 . Similarly, for 0 < α < π,
mα(λ) = cot α + ( √ −λ sin 2 α) −1 + o(|λ| −1/2 )
as λ → ∞ along any non-real ray.
By a non-real ray we always mean a half-line starting at the origin
which is not part of the real line. Here and later the square root is
always the principal branch, i.e., the branch with a positive real part
Now note that, up to constant multiples, the Weyl solution ψ is
determined by the boundary condition at b. For α = 0 we have
ψ ′ (0, λ)/ψ(0, λ) = m0(λ), so keeping a fixed boundary condition at b
we obtain m0(λ) = (sin α+mα(λ) cos α)/(cos α−mα(λ) sin α). Solving
for mα gives
mα(λ) = cos α m0(λ) − sin α
sin α m0(λ) + cos α
= cot α − (m0(λ) sin 2 α) −1 +
cos α
m0(λ) sin 2 α(m0(λ) sin α + cos α) .
Thus, the formula for m0 immediately implies that for mα, 0 < α < π,
so that we only have to prove the formula for m0. This will require
good asymptotic estimates of the solutions ϕ and θ.
Lemma 12.3. If u solves −u ′′ + qu = λu with fixed initial data in 0
one has
(12.4) u(x) = u(0)(cosh(x √ −λ) + O(1)(e x
0 |q|/√ |λ| − 1)e x √ −λ )
uniformly in x, λ.
+ u′ (0)
√ −λ (sinh(x √ −λ) + O(1)(e x
0 |q|/√ |λ| − 1)e x √ −λ ),
Proof. Solving the equation u ′′ + λu = f and then replacing f by
qu gives
(12.5) u(x) = cosh(kx)u(0) + sinh(kx)
k
+
u ′ (0)
x
where we have written k for √ −λ. Setting
0
sinh(k(x − t))
q(t)u(t) dt,
k
g(x) = |u(x) − cosh(kx)u(0) − sinh(kx)
u
k
′ −x Re k
(0)|e
1 If g is a positive function the notation f(λ) = o(g(λ)) as λ → ∞ means
f(λ)/g(λ) → 0 as λ → ∞.