Multiplicative superadditivity and dimension-free Mourre bands for the Molchanov–Vainberg Laplacian

Abstract

Let D=∏ᵢ₌₁ᵈΔᵢ be the Molchanov–Vainberg Laplacian on ℓ²(ℤᵈ), Δᵢ=(Sᵢ+Sᵢ^(*))/2. For the conjugate operators A=Σⱼρ_(jκ)A_(jκ) used by Golénia and Mandich, a strict Mourre estimate at an energy E is equivalent to strict positivity of an explicit Chebyshev polynomial G_κ on the constant-energy surface S_E={x ∈ [-1,1]ᵈ:∏ₗxₗ=E}. Golénia and Mandich proved that G_κ|_(S_E) is separable — G_κ|_(S_E)(x)=EΣᵢP(xᵢ²) for a single-variable P with P(1)=0 — and deduced from it that raising the dimension can only shrink the set of good energies. Nothing was known about the converse, and Mandich, who treats d=2 throughout, re-runs his d=3 numerics with the two-dimensional coefficients and conjectures that they still work, calling the phenomenon "quite mysterious and surprising". We isolate the property that governs the converse: P is multiplicatively superadditive on a region, P(uv) ≤ P(u)+P(v). When it holds, the minimum of ΣᵢP(uᵢ) under ∏ᵢuᵢ=z equals P(z) in every dimension, and is attained at (z,1,…,1). For κ=4 and Mandich's first band — Σ={4,8}, ρ₈=(17+8sqrt5)/62 — we prove multiplicative superadditivity on [1/10,1] and conclude, for every d ≥ 1 and every E ∈ (0,1] with E² ≥ 1/10, that min_(S_E)G₄=E P(E²), a number in which d does not appear. Hence G₄|_(S_E)>0 holds in one dimension if and only if it holds in every other; inside that range the positivity set is exactly (E₁,cos(π/4)) ∪ (sqrt(E₁),1), E₁=(sqrt5-1)/2, in all dimensions at once. In particular the n=1 case of Mandich's Conjecture 1.15 holds in every dimension, at the level of strict positivity to which his own equivalence reduces it, with a sharp and attained constant. The same works for his second band, Σ={4,8,12,28}, whose coefficients he gives exactly in ℚ(sqrt5): its degree-14 polynomial P₂ is multiplicatively superadditive on [1/3,1], so min_(S_E)G^((2))₄=E P₂(E²) whenever E² ≥ 1/3, in every dimension, G^((2))₄ being the combination those coefficients build. The n=2 case of the conjecture follows in the same sense, and at d=2 that band, certified until now on 93.40 identity dividing out the double zero forced by P(1)=0, followed by exact decompositions over ℤ[E₁] at the interior zeros of the kernel — one for the first band, two for the second — and integer Bernstein certificates. The two-sided band set is genuinely dimension-dependent, so the results are stated — as in Mandich — for strict positivity. All polynomial statements are machine-checked in Lean 4; the passage from them to the Mourre estimate itself is the sources' functional calculus, quoted rather than reproved.

Open review

This founding-collection manuscript received AI review before publication. Independent human review is open. Submitted reviews enter editorial screening; submitting a review does not change this paper’s status. Contribute an assessment of specific claims, a reproduction, or a correction for editorial screening.

Archived files

1. Version 2 · current (opens in a new tab)

   Source snapshot 2026-09-07 03:53 UTC

   File fingerprint

   8d58dc511c026d95080eab7e990dfeb21601504f05e131f5616ffd51b81dccf8