On The Spectral Properties of Discrete Landscape Functions

Abstract

We study the landscape function on a discretized interval. The discrete landscape has an explicit closed form, and its spectral coefficients can be computed exactly. We show that these coefficients lie in explicit abelian extensions of $\mathbb{Q}$, and obtain the bound $[\mathbb{Q}(c_k(N)):\mathbb{Q}]\le \phi (N)$ for every odd $k$. For the first coefficient, exact computation for $3\le N\le 30$ gives the full degree $\phi (N)$ in every case, motivating the Chandra--Jain conjecture that $[\mathbb{Q}(c_1(N)):\mathbb{Q}]=\phi (N)$ for all $N\ge 3$. We then reduce the higher modes to the coprime case and discuss the remaining degree question. We also discuss connections with parity, the Arnold cat map, and Lefschetz numbers. The Chandra--Jain conjecture has since been proved by Q. Zhou (Zenodo, doi:10.5281/zenodo.21935814).

v2: Named conjecture in abstract; added note on proof of Conjecture 4.5 by Q. Zhou (Zenodo:10.5281/zenodo.21935814). 11 pages, 1 figure