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Abstract This paper investigates uniqueness results for perturbed periodic Schrödinger operators on

Z^{d}

. Specifically, we consider operators of the form

H = - Δ + V + v

, where Δ is the discrete Laplacian,

V : Z^{d} \to R

is a periodic potential, and

v : Z^{d} \to C

represents a decaying impurity. We establish quantitative conditions under which the equation

- Δ u + V u + v u = λ u

, for

λ \in C

, admits only the trivial solution

u \equiv 0

. Key applications include the absence of embedded eigenvalues for operators with impurities decaying faster than any exponential function and the determination of sharp decay rates for eigenfunctions. Our findings extend previous works by providing precise decay conditions for impurities and analyzing different spectral regimes ofλ.

Bergman metrics with constant holomorphic sectional curvatures

https://doi.org/10.1515/crelle-2025-0013

Huang, Xiaojun; Li, Song-Ying; Treuer, John N (March 2025, Journal für die reine und angewandte Mathematik (Crelles Journal))

Abstract The paper studies complex manifolds whose Bergman metrics are incomplete but have constant holomorphic sectional curvature.We will construct a real analytic unbounded domain in

C^{2}

\mathbb{C}^{2}whose Bergman metric is well-defined and has a positive constant holomorphic sectional curvature, which appears to be the first example of this kind.We will answer a long standing folklore conjecture that a Stein manifold has a negative constant holomorphic sectional curvature if and only if it is biholomorphic to a ball with a pluripolar set removed.Together with the uniqueness of a moment problem in the appendix of the paper provided by John Treuer, we will show that, under natural assumptions, there is no complex manifold whose Bergman metric is flat.

Free, publicly-accessible full text available March 22, 2026

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