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This paper investigates the rates of decay at infinity for solutions to variable-coefficient Schrödinger equations of the form -div(A∇u) + W · ∇u + Vu = λu in cylinders Tᵈ × ℝᵐ. The authors establish optimal decay rates that depend on the decay of ∇A at infinity, particularly focusing on the condition |∇A(x)| ≤ L|x|⁻¹ for large |x|. They also construct explicit solutions to demonstrate the sharpness of these estimates.

Proving optimal rates of decay at infinity for solutions to variable-coefficient Schrödinger equations.
Establishing that the condition |∇A(x)| ≤ L|x|⁻¹ at infinity is the sharp threshold for unique continuation at infinity with a polynomial power.
Constructing explicit examples of solutions that demonstrate the sharpness of the derived decay estimates.
Analyzing the impact of the decay rate of |∇A(x)| on the decay of solutions, particularly for polynomial decay.

The paper employs a combination of analytical techniques to establish its results. The core methodology involves:

Frequency Functions and Monotonicity: The authors introduce and analyze generalized frequency functions, similar to those used in local unique continuation problems. These functions are shown to be almost monotone, which leads to three-cylinder inequalities.
Three-Cylinder Inequalities: These inequalities provide quantitative relationships between the size of a solution in nested balls (or cylinders in this context). They are crucial for passing information about the solution from one region to another, particularly for studying behavior at infinity.
Iterative Argument: A general iterative scheme is developed to establish Landis-type results. This scheme precisely links the rate of decay of |∇A(x)| to the rate of decay of solutions at infinity.
Construction of Sharp Examples: The paper constructs explicit solutions to demonstrate that the derived decay rates are indeed optimal. These constructions often involve adapting techniques from previous works, such as those by Miller and [KLP25].

Equation: The primary equations considered are generalized Schrödinger equations of the form:
- -div(A∇u) + W · ∇u + Vu = λu in Tᵈ × ℝᵐ
- -div(A∇u) = λu in Tᵈ × ℝᵐ
Matrix Assumptions: The coefficient matrix A is assumed to be uniformly elliptic, symmetric, and locally Lipschitz, with a specific block structure: A(θ, x) = (A¹(θ,x) A¹²(θ,x)) (A²¹(θ,x) A²(θ,x)) where A¹ is d×d and A² is m×m.
Decay Conditions on ∇A: The analysis heavily relies on assumptions about the decay of |∇A(x)| at infinity. Key conditions include:
- |∇A(x)| ≤ L|x|⁻¹ for τ ∈ [0,1] (Theorem A)
- |∇A(x)| ≤ L|x|⁻¹ (Theorem B)
- |∇A(x)| ≤ L/(1+|x|log(|x|+1)) (Theorem C)
Solution Properties: Solutions u are assumed to be normalized and have bounded growth at infinity.
Key Quantities:
- Frequency function N(r).
- Monotonicity functions H(r), D(r), L(r), I(r), J(r).
- Decay rate parameters τ, p, q, κ, ε.
- Integral quantities like φ(r) = ∫ η(s) ds.

Theorem A: Establishes decay rates for solutions based on the polynomial decay of |∇A(x)|, specifically |∇A(x)| ≤ L|x|⁻ᵗ for τ ∈ [0,1]. The decay rate of the solution depends on τ and L. For τ ∈ [0,1), solutions cannot decay faster than exp{-C exp(|x|¹⁻ᵗ)}. For τ = 1, solutions cannot decay faster than exp{-C|x|ᵖ} for some p > 0.
Theorem B: Provides sharp examples demonstrating that the condition |∇A(x)| ≤ L|x|⁻¹ at infinity is necessary for unique continuation at infinity with a polynomial power.
Theorem C: Shows that if |∇A(x)| decays slightly faster than |x|⁻¹, specifically |∇A(x)| ≤ L/(1+|x|log(|x|+1)), then eigenfunctions can decay faster than polynomial decay, like exp{-C|x| exp(log(|x|)³/⁴)}.
Quantitative Unique Continuation: The paper provides quantitative unique continuation results at infinity, linking the decay rate of |∇A| to the decay rate of solutions.

The paper notes that it is unclear whether a result similar to Theorem 3.5 (faster decay of eigenfunctions) holds if the decay of |∇A| is only |∇A(x)| ≤ L₂/(1+|x|) for small L₂, due to obstructions related to the critical exponent q.
The construction of sharp examples in Section 4 is focused on the 3-dimensional domain T² × ℝ.

Method evidence: "We apply the three-cylinder inequality in (2.30) to solutions of the eigenvalue equation (3.1) when |∇A| exhibits polynomial decay in the x-direction: |∇A(0,x)| ≤ L/(1+|x|)ᵗ for a.e. (0,x) ∈ Tᵈ × ℝᵐ, and some τ ∈ [0,1]." (Section: 3. Applications to Unique Continuation at Infinity)
Result evidence: "Theorem A (Rate of decay at infinity when coefficients have polynomial decay). Let A be a uniformly elliptic, symmetric, bounded matrix that has the block structure in (1.3) and satisfies (1.4). Let W ∈ L∞(Td × Rm; Cd+m), V ∈ L∞(Td × Rm; C), λ∈ C, and define q := 2 if W≠0, q := 4/3 if W=0, V≠0, and q := 1 otherwise. Let u be a normalized solution of – div(Au) + W · ∇u + Vu = λu in Td × Rm with bounded growth at infinity. (a) If τ∈ [0, 1), then there exists Ro ≫ 1 and C > 0 so that whenever |xo| ≥ Ro, it holds that ||u||L2(TdxB1(xo)) ≥ exp {- exp (C|xo|¹⁻ᵗ)}." (Section: 1.1. Theorem statements)
Limitation evidence: "It is unclear whether such a result is true, and at least with the techniques of this paper (namely, the technology of Proposition 3.1), it does not seem possible to prove. The main obstruction is that the critical exponent q from (3.26) is not strictly larger than 1 when W = V = 0. We elaborate on this below." (Section: 3.3. Unique continuation for additional decay)

Optimal rates of decay at infinity for solutions to Schrödinger equations

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