Spectral Estimates and Non-Selfadjoint Perturbations of Spheroidal Wave Operators

We derive a spectral representation for the oblate spheroidal wave operator which is holomorphic in the aspherical parameter $\Omega$ in a neighborhood of the real line. For real $\Omega$, estimates are derived for all eigenvalue gaps uniformly in $\Omega$. The proof of the gap estimates is based on detailed estimates for complex solutions of the Riccati equation. The spectral representation for complex $\Omega$ is derived using the theory of slightly non-selfadjoint perturbations.Comment: 33 pages, LaTeX, 3 figures, typo in Lemma 4.1 corrected (published version

Decoupling of Neoarchean sulfur sources recorded in Algoma-type banded iron formation

Highlights • The genetic model for Algoma-type BIF is modified taking into account S-MIF results. • Metal and sulfur sources are decoupled and reflect diverse microbial metabolisms. • Sulfur deposited with oxide-facies BIF is mostly atmospheric in origin. • Little juvenile sulfur is found, despite the proximity to volcanic sources. Abstract Neoarchean Algoma-type banded iron formations (BIFs) are widely viewed as direct chemical precipitates from proximal volcanic–hydrothermal vents. However, a systematic multiple sulfur isotope study of oxide-facies BIF from a type locality in the ca. 2.74 Ga Temagami greenstone belt reveals mainly bacterial turnover of atmospheric elemental sulfur in the host basin rather than deposition of hydrothermally cycled seawater sulfate or sulfur from direct volcanic input. Trace amounts of chromium reducible sulfur that were extracted for quadruple sulfur isotope (32S–33S–34S–36S) analysis record the previously known mass-independent fractionation of volcanic SO2 in the Archean atmosphere (S-MIF) and biological sulfur cycling but only minor contributions from juvenile sulfur, despite the proximity of volcanic sources. We show that the dominant bacterial metabolisms were iron reduction and sulfur disproportionation, and not sulfate reduction, consistent with limited availability of organic matter and the abundant ferric iron deposited as Fe(OH)3. That sulfur contained in the BIF was not a direct volcanic–hydrothermal input, as expected, changes the view of an important archive of the Neoarchean sulfur cycle in which the available sulfur pools were strongly decoupled and only species produced photochemically under anoxic atmospheric conditions were deposited in the BIF-forming environment