Landis-type results for discrete equations

TL;DR

The paper introduces Landis-type uniqueness results for two discrete-settings: the semidiscrete heat equation on $(h\mathbb{Z})^d$ and the stationary discrete Schrödinger equation on the same lattice with bounded potentials. It develops a unified framework of quantitative upper and lower bounds by combining weighted-energy methods, log-convexity, and Carleman inequalities, revealing an interpolation between continuum-like and discrete behaviors as the mesh size $h$ and scale $R$ vary. For the heat equation, two-time decay conditions force trivial solutions, with separate near-continuum and purely discrete regimes; for the elliptic equation, exponential decay at infinity yields trivial solutions, with exponents echoing the continuum $4/3$-type and discrete decay patterns. The results illuminate how discretization degrades or alters decay rates, provide near-optimal bounds in many regimes, and open questions about sharpness and extensions to non-stationary discrete dynamics. Overall, the work advances Landis-type uniqueness in a discretized setting, connecting discrete and continuum theories and quantifying the scale-dependent behavior of decaying solutions.

Abstract

We prove Landis-type results for both the semidiscrete heat and the stationary discrete Schrödinger equations. For the semidiscrete heat equation we show that, under the assumption of two-time spatial decay conditions on the solution $u$, then necessarily $u\equiv 0$. For the stationary discrete Schrödinger equation we deduce that, under a vanishing condition at infinity on the solution $u$, then $u\equiv 0$. In order to obtain such results, we demonstrate suitable quantitative upper and lower estimates for the $L^2$-norm of the solution within a spatial lattice $(h\mathbb{Z})^d$. These estimates manifest an interpolation phenomenon between continuum and discrete scales, showing that close-to-continuum and purely discrete regimes are different in nature.

Figures (1)

• Figure 1: Illustration in two dimensions; for a point in the ball of center $(0,0)$ and radius $R$ and the metric $|\cdot|_{\infty}$, say $(j,R)$ with $j,R\in h\mathbb{Z}$, $j<R$, and for a solution of the equation, then $u(j,R-h)= 4u(j,R)-u(j-h,R)-u(j+h,R)-u(j,R+h)+Vu(j,R)$ holds. Observe that on the right hand side of this equation there are $2\cdot 2+(2\cdot 2-2)$ evaluations on the ball of radius $R$ ($2\cdot 2$ are coming from the values of $u(j,R)$ and $(2\cdot 2-2)$ from the values of $u(j-h,R), u(j+h,R)$), and one evaluation of $u$ on the ball of radius $R+h$. In higher dimensions, this corresponds to $2d + (2d-2)$ evaluations on the ball of radius $R$ and one on the ball of radius $R+h$.

Theorems & Definitions (45)

• Theorem 1: Landis-type result for the semidiscrete heat equation
• Theorem 2: Landis-type result for the stationary discrete Schrödinger equation
• Theorem 1: Upper bound, close-to-continuum regime
• Remark 1.1
• Theorem 2: Upper bound, purely discrete regime
• Remark 1.2
• Remark 1.3
• Theorem 3: Lower bound
• Theorem 4: Lower bound
• Remark 1.4