A hard boundary in four dimensions
An arXiv version 1 preprint dated 26 August 2026 reports uniform bounds for the standard discrete Laplacian H0 on the lattice Z^d, covering dimensions d ≥ 4. The work asks how far its resolvent, the inverse R0(z) = (H0 - z)^-1, can carry a sequence in ℓ^p to one in the conjugate space ℓ^{p′}, both globally and away from threshold energies. This is a mathematical operator problem, not a study based on human or animal observations.
In four dimensions, the main unweighted bound holds for 1 ≤ p < 4/3, with the output exponent p′. At the limiting power p = 4/3, the stated estimate comes with a square-root logarithmic loss. That means the endpoint carries an extra factor that grows slowly, rather than giving a lossless endpoint estimate. The power range is described as sharp, but the author does not assert that the logarithmic loss itself is necessary.
The geometry behind the bound
The proof starts with local Fourier decay, meaning how rapidly a frequency-space transform falls away from a localized part of the Fermi surface. The surface’s local shape determines which oscillatory estimate applies. At regular points, directions whose quadratic terms vanish are treated as cubic phases, while the remaining directions are treated as quadratic phases.
This produces three decay patterns. In the low-flat regular case, the bound is a power law modified by a logarithm. For m ≥ 5, the pure-power exponent is (d - m)/2 + (m - 1)/3. At a critical point, the exponent is (d - 2)/2. The energies λ = 0 and λ = 4d are exceptional in a more basic way: there is no local regular part of dimension d - 1 there.
Those local estimates are converted into weighted resolvent bounds on regular patches. At nondegenerate critical points, the proof switches to stationary phase, together with an endpoint Strichartz argument. The result is a set of estimates split according to the geometry of the relevant part of the spectrum.
Different dimensions, different ranges
For d ≥ 5, the global unweighted range is 1 ≤ p ≤ 2(d + 2)/(d + 5). The corresponding weighted range is 1 ≤ r ≤ 2(d + 2)/3. These are uniform bounds for the global setting covered by the theorem.
The picture changes away from threshold energies. If the spectral parameter is kept a fixed positive distance δ from the threshold set, odd dimensions d ≥ 5 retain the global range. Even dimensions d ≥ 6 receive an improved unweighted range, 1 ≤ p ≤ 2(2d + 5)/(2d + 11), and an improved weighted range, 1 ≤ r ≤ (2d + 5)/3. The improvement is conditional on separation from the thresholds and depends on parity.
A test of how far the estimates can go
The paper also constructs obstructions to extending these ranges. Its test functions occupy anisotropic frequency boxes, with normal scale ε, cubic scale ε^(1/3), and quadratic scale ε^(1/2). Their different side lengths reflect the phase geometry, and the resulting volumes force restrictions on the allowed p exponents.
Globally, the construction makes p ≤ 2(d + 2)/(d + 5) necessary. Away from thresholds, it gives the same necessary condition for odd d ≥ 5 and p ≤ 2(2d + 5)/(2d + 11) for even d ≥ 4. In four dimensions, the power range is sharp, but the construction does not settle whether the square-root logarithmic endpoint loss is genuinely required.
Results that follow from the main bounds
The estimates are then used for consequences involving perturbed lattice operators. A Birman–Schwinger bound holds for potentials V in ℓ^s(Z^d) when d = 4 and 1 ≤ s < 2, or when d ≥ 5 and 1 ≤ s ≤ (d + 2)/3. For real V, the paper states that H0 + γV is unitarily equivalent to H0 for sufficiently small real γ.
A related Kato smoothing result is stated for W in ℓ^r(Z^d), with 1 ≤ r < 4 in d = 4 and 1 ≤ r ≤ 2(d + 2)/3 in d ≥ 5, together with a spacetime ℓ^2 estimate. In another application, for d ≥ 5 and s0 = (d + 2)/3, complex eigenvalues z outside [0, 4d] obey a distance-dependent restriction when s > s0 and V belongs to ℓ^s; at s = s0, a sufficiently small ℓ^s0 norm excludes such eigenvalues.
The paper also gives a thin spectral projection estimate. For an interval I lying compactly inside (0, 4d) and away from E_d, every q > q_d* yields a bound of order α^(1/2), with q_d* = 4 in d = 4 and q_d* = 2(2d + 5)/(2d - 1) in d ≥ 5. This result is restricted to intervals away from the stated exceptional energies.
The unresolved edge
The most important caveat is the four-dimensional endpoint. The preprint gives the range below 4/3 and an endpoint estimate with a square-root logarithmic loss, but it does not claim that the loss is necessary. Whether a lossless ℓ^(4/3)(Z^4)-to-ℓ^4(Z^4) estimate holds remains open in the supplied analysis.
The conclusions concern the free discrete Laplacian on Z^d. Away-from-threshold improvements require fixed positive separation from the threshold set. The document is an arXiv version 1 preprint dated 26 August 2026.
Paper data and sources
Original title: Uniform Resolvent Estimates for the Discrete Schrödinger Operator in Higher Dimensions
Authors: Yuda Chen
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text