Volume 120, Number 1 (1994), in the article “Lower bounds for the spectra of the Schrödinger operators with magnetic fields,” by Naomasa Ueki, pages 344–379.

The proof of Lemma 5.1 is incorrect, since (5.9) should be corrected as

(1)  ∥e^{−L^{n}(b)}∥_{End(L^{2}(M))} = sup_{\begin{matrix}φ∈C_{0}(M) \\ ∥φ∥_{L^{2}(M)}=1 \end{matrix}} ∥e^{−L^{n}(b)}φ∥_{L^{2}(M)}.

This mistake affects all the mechanism of the proof of Section 5.

However, if the manifold M is compact, then the results of Section 5 hold true. Theorem 5.1’. If the manifold M is compact, then

(2)  \underline{lim}_{ξ→∞} \frac{inf spec L(ξb)}{ξ} ≥ \frac{1}{4} inf_{x∈M} ∥db(x)∥_{1}.

Proof. We take δ > 0 arbitrarily. Since M is compact, we can take a finite open covering {D_{ι}}_{ι=1}^{N} of M and a family of C^{2} functions {h_{ι}}_{ι=1}^{N} satisfying the following:

(3)

(i) for each ι, the diameter of D_{ι} is less than δ and, on \overline{D_{ι}}, we can take subbundles U^{ι}, V^{ι}, W^{ι} and an almost complex structure J^{ι} so that (2.2) holds and B^{ι}(x_{ι}) = ∥db(x_{ι})∥_{1}/4 for some x_{ι} ∈ D_{ι}, where B^{ι} is the corresponding B defined in (2.4), (ii) supph_{ι} ⊂ D_{ι} and ∑_{ι}(h_{ι})^{2} ≡ 1.

Then we obtain

(4)  L(ξb) = ∑_{ι} h_{ι}L(ξb)h_{ι} − \frac{1}{2} ∑_{ι} ∥dh_{ι}∥^{2}

(cf. B.Simon, Ann.Inst.Henri Poincaré, Physique Théorique, 38 (1983), 295–307). Noting

(5)  sup_{x∈M} ∑_{ι} ∥dh_{ι}(x)∥^{2} < ∞,

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we estimate as

(6)  inf spec L(ξb) ≥ inf_{ι} inf spec L(ξb)_{D_{ι}} − \frac{1}{2} sup_{x∈M} ∑_{ι} ∥dh_{ι}(x)∥^{2},

where L(ξb)_{D_{ι}} is the corresponding L(ξb) on \overline{D_{ι}}. By applying Theorem 2.1, we have

(7)  inf spec L(ξb)_{D_{ι}} ≥ − sup_{y∈D_{ι}} ∑_{j=1}^{3} R_{j}^{ι}(y) + ξ inf_{y∈D_{ι}} B^{ι}(y),

where R_{j}^{ι} are the corresponding R_{j} defined in (2.3). Therefore we obtain

(8)  \underline{lim}_{ξ↗∞} \frac{inf spec L(ξb)}{ξ} ≥ inf_{ι} inf_{y∈D_{ι}} B^{ι}(y).

Since δ was arbitrary, we obtain (2).

The derivation of Corollaries 1 and 2 from Theorem 5.1 is correct. This proof is valid more generally under the situation that, for any δ > 0, we can take a locally finite open covering {D_{ι}} and a family of C^{2} functions {h_{ι}} satisfying (3) and (5), and the assumptions (A.2), (A.3) hold.