Aryaman Chandra | Searcharxiv

arXiv2026

On The Spectral Properties of Discrete Landscape Functions

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 $[\mathbb{Q}(c_k(N)):\mathbb{Q}]\leq\varphi(N)$ for every odd $k$. For the first coefficient, exact computation for $3\leq N\leq 30$ gives the full degree $\varphi(N)$ in every case, motivating the Chandra--Jain conjecture that $[\mathbb{Q}(c_1(N)):\mathbb{Q}]=\varphi(N)$ for all $N\geq 3$. 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).

math.DS↗

arXiv2026

Arithmetic Landscape Functions of a Discrete Cat Map

We study the diagonal Green function $\widetilde{u}(x)=[L_N^{-1}]_{x,x}$ of the operator $L_N=I-\alpha P$ on the finite torus $(\mathbb{Z}/N\mathbb{Z})^2$, where $P$ is the transfer operator of the discrete cat map $T_N(x)=Ax \bmod N$. We prove the exact formula $\widetilde{u}(x)=(1-\alpha^{k_x})^{-1}$, where $k_x$ is the minimal period of $x$ under $T_N$. This formula appears to be new. It shows that the diagonal landscape is a complete spectral invariant of the orbit structure, depending on each point only through its orbit length. Since $\det(A-I)=-1$ is a unit in $\mathbb{Z}/N\mathbb{Z}$ for every $N\ge2$, the origin is the unique fixed point of $T_N$ and the unique global maximum of $\widetilde{u}$. The resulting localization is driven by arithmetic alone, with no disorder and no broken symmetry, a mechanism distinct from classical Anderson theory and from Filoche--Mayboroda landscape theory. We further establish the Chandra Green--Zeta Identity, showing that the Green trace satisfies $\operatorname{tr}(G_N)=N^2-\alpha\frac{d}{d\alpha}\log Z_N(\alpha)$, where $Z_N$ is the dynamical zeta function of $T_N$, and that a Laplacian perturbation degrades the localization gap at first order in $\varepsilon$. All results are verified computationally.

math.DS↗