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Wen-Cai Liu

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Preprint Sep 2026

Sharp decay thresholds for eigenvalues of discrete Schr\"odinger operators on $\mathbb{Z}^2$

We study eigenvalues of discrete Schr\"odinger operators $H=\Delta+V$ on $\mathbb{Z}^2$, where $\Delta$ is the uncentered Laplacian, i.e., the un-normalized adjacency operator of $\mathbb{Z}^2$ and $V$ decays at infinity. By Weyl's theorem, the essential spectrum of $H$ is $[-4,4]$. We determine the sharp decay thresho...

S. Ganguly, Wen-Cai Liu · 0 citations

Preprint Sep 2026

Irreducibility of the Bloch variety for periodic Schr\"odinger operators in arbitrary dimension

Let $d\ge1$ and $V\in C(\mathbb{T}^d;\mathbb{C})$. We prove that the Bloch variety of $-\Delta+V$ is irreducible modulo periodicity. The proof combines holomorphic continuation of spectral band functions with integer translations of quasimomentum. Using analytic geometry and perturbation estimates along imaginary rays,...

Wen-Cai Liu · 0 citations