Correction to: Upper bounds for the resonance counting function

of Schr � odinger operators in odd dimensions

Richard Froese Department of Mathematics University of British Columbia Vancouver, British Columbia, Canada

The proof of Lemma 3.4 relies on the incorrect equality μ_{j}(AB) = μ_{j}(BA) for singular values (for a counterexample, see [S], p. 4.) Thus, Theorem 3.1 as stated has not been proven. However, with minor changes, we can obtain a bound for the counting function in terms of the growth of the Fourier transform of |V |. The author thanks Barry Simon for pointing out this error.

Here is the corrected version of Theorem 3.1. Theorem Suppose that V is a super-exponentially decaying potential with

ˆ|V |(z) ≤ Ce^{Φ(|z|)}

for a positive, increasing function Φ. Then

n(r) ≤ CΦ^{n}(cr) + O(Φ^{n−1}(cr))

for some constants c and C.

These are the changes needed to prove the bound for |ϕ(k)| in Lemma 3.4. Using det(1+AB) = det(1 + BA) and Fan’s inequality μ_{n+m+1}(AB) ≤ μ_{n+1}(A)μ_{m+1}(B) (see [S]) we arrive at

μ_{j}(T (k)) ≤ C|k|^{n−2}μ_{[(j+1)/2]}(F_{V}^{T} (−k))μ_{[(j+1)/2]}(F_{|V |}(−k))  ()

where [·] denotes the integer part. Now

μ_{[(j+1)/2]}(F_{|V |}(−k)) = (μ_{[(j+1)/2]}V_{k}))^{1/2}

where this time V_{k} is the integral operator with integral kernel ˆ|V |(\overline{k}ω − kω′). We then obtain the bound

μ_{[(j+1)/2]}(F_{|V |}(−k)) ≤ Ce^{(Φ−δ[(j+1)/2]^{(1/(n−1)})/2}

and the same bound for μ_{[(j+1)/2]}(F_{V}^{T} (−k)). This leads to

μ_{j}(T (k)) ≤ Ce^{Φ−δ′j^{1/(n−1)}}

where Φ = Φ((2 + ϵ)|k|) for some ϵ > 0 and δ′ = δ2^{−1/(n−1)}. The rest of the proof is identical.