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Proof. By Proposition 5.14 the operator ∂α is semi-Fredholm, so it is enough to prove that its
kernel and co-kernel are both (0).
Let u ∈ X 1,p (Σ) be an element of the kernel of ∂α. By Proposition 5.12, u(z) has exponential
decay for |Re z| → +∞ together with all its derivatives. By Lemma 5.15, u is smooth on Σ \ {0},
it is continuous at 0, and Du(z) = O(|z| −1/2 ) for z → 0. Then the function w := u 2 belongs to
W 1,q (Σ,�) for every q < 4. Moreover, w is real on the boundary of Σ, and it satisfies the equation
∂w + 2αw = 0.
Since 0 < 2α < π, e 2αiÊ∩Ê=(0), so the assumptions of Theorem 5.11 (ii) are satisfied, and the
operator
∂2α : W 1,q
Ê,Ê(Σ,�) → L q (Σ,�)
is an isomorphism. Therefore w = 0, hence u = 0, proving that the operator ∂α has vanishing
kernel.
Let v ∈ X q (Σ), 1/p + 1/q = 1, be an element of the cokernel of ∂α. By Lemma 5.16, v is
smooth on Σ \ {0}, v(s) ∈ iÊfor s < 0, v(s) ∈Êfor s > 0, v solves ∂v − αv = 0, and the function
w(ζ) := 2ζv(ζ 2 ) (104)
is smooth in Cl(À+ ) ∩�1 and real on the boundary ofÀ+ . In particular, v(z) = O(|z| −1/2 ) and
Dv(z) = O(|z| −3/2 ) for z → 0. Furthermore, by Proposition 5.12, v and Dv decay exponentially
for |Re z| → +∞. More precisely, since the spectrum of the operator Lα on L 2 (]0, 1[,�),
domLα = W 1,2
Ê,Ê(]0, 1[,�) = � u ∈ W 1,2 (]0, 1[,�) | u(0), u(1) ∈Ê�
, Lα = i d
+ α,
dt
is α + π�, we have min σ(Lα) ∩ [0, +∞) = α, hence
|v(z)| ≤ ce −α|Re z| , for |Re z| ≥ 1. (105)
If w(0) = 0, the function v vanishes at 0, and Dv(z) = O(|z| −1 ) for z → 0, so v2 belongs to
W 1,q
Ê,Ê(Σ,�) for any q < 2, it solves ∂v2 −2αv2 = 0, and as before we deduce that v = 0. Therefore,
we can assume that the real number w(0) is not zero.
Consider the function
Since ∂v = αv,
f : Σ \ {0} →�, f(z) := e −αz/2 v(z).
∂f(z) = −αe −αz/2 v(z) + e −αz/2 ∂v(z) = e −αz/2 (−αv(z) + αv(z)) = 0,
so f is holomorphic on the interior of Σ. Moreover, f is smooth on Σ \ {0}, and
f(s) = e −αs/2 v(s) ∈ iÊfor s < 0, f(s) = e −αs/2 v(s) ∈Êfor s > 0, (106)
f(s + i) = e −αs/2 v(s + i)e αi/2 ∈ e αi/2Êfor every s ∈Ê. (107)
Denote by √ z the determination of the square root on�\Ê− such that √ z is real and positive
for z real and positive, so that √ z = √ z. By (104),
Finally, by (105),
−αz/2 1
f(z) = −e √ w(
z √ z) = w(0)
√ + o(|z|
z −1/2 ) for z → 0. (108)
lim f(z) = 0. (109)
|Re z|→+∞
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