Correction to: Spectral analysis of the diffusion operator with random jumps from the boundary
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Mathematische Zeitschrift
CORRECTION
Correction to: Spectral analysis of the diffusion operator with random jumps from the boundary Martin Kolb1 · David Krejˇciˇrík2 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Correction to: Math. Z. (2016) 284:877–900 https://doi.org/10.1007/s00209-016-1677-y Abstract We correct a wrong statement in the original article that the studied operator is quasi-accretive. In fact, in this corrigendum we show that the numerical range of the operator coincides with the whole complex plane. We argue that the other statements in the original article still hold. In the original article we studied the operator H in L 2 ((− π2 , π2 )) defined by H ψ := −ψ , ψ ∈ D(H ) := ψ ∈ H 2 − π2 , π2 ψ − π2 = ψ π2 a = ψ π2 . On page 880 in the original article we presented a proof that H is quasi-accretive, but an additional derivative was missing on the second line of the proof, making the total argument irreparable.1 The correct identity reads π 1 2 2 (ψ, H ψ) = ψ − |ψ|2 (x) d x π 2 − 2
and it follows that the quasi-accretivity actually does not hold. In fact, the whole complex plane belongs to the numerical range. Proposition {(ψ, H ψ) | ψ ∈ D(H ), ψ = 1} = C. Proof 2 We employ the identity (ψ, H ψ) = ψ 2 − ψ¯
π π ψ 2 − ψ − π2 , 2
1 We are grateful to Matˇej Tušek for pointing out the mistake to us. 2 The present proof is based on an initial idea of Matˇej Tušek.
The original article can be found online at https://doi.org/10.1007/s00209-016-1677-y.
B
David Krejˇciˇrík [email protected]
1
Department of Mathematics, Universität Paderborn, Warburger Str. 100, 33098 Paderborn, Germany
2
Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, 12000 Prague 2, Czech Republic
123
M. Kolb, D. Krejˇciˇrík
which follows by integrating by parts and using the boundary conditions. Let φ ∈ C ∞ (R) be an arbitrary complex-valued function such that its support is contained in ( π2 a, π2 ]. We set
ψε (x) :=
1+ 1
√
εφ
π 2
− ε −1
π 2
−x
if x ∈ π2 bε , π2 , otherwise,
with bε := 1 − ε(1 − a) and any ε ∈ (0, 1]. Clearly, ψε ∈ D(H ) and a straightforward calculation yields φ π2 2 (ψε , H ψε ) = φ − √ , ε 2 3/2 ψε = π + 2 ε φ(y) dy + ε 2 |φ(y)|2 dy −−→ π. R
R
ε→0
Since φ ( π2 ) is an arbitrary complex number, the desired claim follows by sending ε to zero and recalling the convexity of the numerical range.
As a consequence of the lack of quasi-accretivity, the operator H is not quasi-m-accretive and the corresponding part of Proposition 3 in the original article does not hold. On the other hand, H is still an operator with compact resolvent. To make the argument of the proof of Proposition 3 in the original article work, it is enough to show that the resolvent of H exists for one complex z. However, given an arbitrary F ∈ L 2 ((− π2 , π2 )), the equation (H − z)ψ = F admits an explicit solution for e
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