Erratum to the paper

“Absolute Continuity of the Spectrum

of a Schrödinger Operator

with a Potential Which is Periodic

in Some Directions and Decays in Others”

Documenta Math Vol. 9 (2004), 107-121

N. Filonov^{1} and F. Klopp^{2}

Received: April 9, 2004

Communicated by Heinz Siedentop

Abstract. Lemma 6.1 and 6.2 in [1] are false as stated there. Below, we correct the proof of Theorem 6.1 accordingly.

2000 Mathematics Subject Classification: 35J10, 35Q40, 81C10

6 One fact from the theory of functions

Lemma 6.1. Let U be an open subset of ℝ^{d}. Let f be a real-analytic function on the set U × (a, b), and pick Λ ⊂ (a, b) such that mes Λ = 0. Then

mes{k ∈ U : ∃λ ∈ Λ s.t. f(k, λ) = 0 and ∂_{k_{1}}f(k, λ) ≠ 0} = 0.  (1)

Proof. The Implicit Function Theorem implies that, for any point (k^{∗}, λ^{∗}) such that f(k^{∗}, λ^{∗}) = 0 ≠ ∂_{k_{1}}f(k, λ^{∗}), we can find rational numbers r̃ > 0, λ̃, a vector k̃ with rational coordinates, and a real analytic function θ defined in Br̃(k̃′, λ̃) such that

1. (k^{∗}, λ^{∗}) ∈ Br̃(k̃, λ̃); 2. θ((k^{∗})′, λ) = k_{1}^{∗};

1N.F.’s research was partially supported by the FNS 2000 “Programme Jeunes Chercheurs”.

2F.K.’s research was partially supported by the program RIAC 160 at Université Paris 13 and by the FNS 2000 “Programme Jeunes Chercheurs”.

Documenta Mathematica 9 (2004) 135–136

3. f(k, λ) = 0 ⇔ θ(k′, λ) = k_{1} if (k, λ) ∈ Br̃(k̃, λ̃).

The Jacobian of the map

(k′, λ) ↦ (θ(k′, λ), k′)

is bounded, so

mes{(θ(k′, λ), k′) : (k′, λ) ∈ Br̃(k̃′, λ̃), λ ∈ Λ} = 0,

and therefore,

mes{k : ∃λ ∈ Λ s.t. (k, λ) ∈ Br̃(k̃, λ̃) and f(k, λ) = 0} = 0.

The set

{(k, λ) : f(k, λ) = 0 and ∂_{k_{1}}f(k, λ) ≠ 0}

can be covered by a countable number of balls Br̃_{i}(k̃_{i}, λ̃_{i}) constructed as above, hence the measure of the set in (1) is also equal to zero.

Theorem 6.1. Let U be a region in ℝ^{d}, Λ be a subset of an interval (a, b) such that mes Λ = 0. Let h be a real-analytic function defined on the set U × (a, b) and suppose that

∀λ ∈ Λ ∃k ∈ U such that h(k, λ) ≠ 0.  (2)

Then,

mes{k ∈ U : ∃λ ∈ Λ s.t. h(k, λ) = 0} = 0.

Proof. The proof of Theorem 6.1 is that given in [1] except that one uses Lemma 6.1.

References

[1] N. Filonov and F. Klopp. Absolute continuity of the spectrum of a Schrödinger operator with a potential which is periodic in some directions and decays in others. Documenta Mathematica, 9:107–121, 2004.

N. Filonov Department of

Mathematical Physics St Petersburg

State University 198504 St Petersburg-Petrodvorets

Russia

filonov@mph.phys.spbu.ru F. Klopp LAGA, Institut Galilée Université Paris-Nord F-93430 Villetaneuse France klopp@math.univ-paris13.fr

Documenta Mathematica 9 (2004) 135–136