Two rigorous Mourre bands for the Molchanov–Vainberg Laplacian in dimension two

Abstract

Let D=Δ₁Δ₂ be the Molchanov–Vainberg Laplacian on ℓ²(ℤ²), with Δᵢ=(Sᵢ+Sᵢ^(*))/2. For a conjugate operator A=Σⱼρ_(jκ)A_(jκ) built from the dilation-type generators A_(jκ), a strict Mourre estimate for D at an energy E is equivalent to strict positivity, on the constant-energy surface S_E={x ∈ [-1,1]²:x₁x₂=E}, of the explicit Chebyshev combination G_κ=Σⱼρ_(jκ)g_(jκ), g_(jκ)(x₁,x₂)=x₂(1-x₁²)U_(jκ-1)(x₁) +x₁(1-x₂²)U_(jκ-1)(x₂). Mandich (arXiv:2201.00410) determines the coefficients ρ_(jκ) band by band, by interpolation at threshold energies, and then supplies plots of G_κ; he states plainly that he was not able to prove a Mourre estimate on any new interval, and that the new bands of absolutely continuous spectrum are "justified mainly by graphical evidence". We turn the first two of those plots, at κ=4 and d=2, into theorems with explicit constants. For the first band (E₁,E₀)=((sqrt5-1)/2,cos(π/4)) and the author's own coefficient ρ₈=(17+8sqrt5)/62, we prove G₄(x₁,x₂) ≥ frac125(E²-tfrac(3-sqrt5)2)(1-2E²), E=x₁x₂, on the whole closed band, the right-hand side vanishing exactly at the two thresholds. For the second band (E₂,E₁)=(1/sqrt3,(sqrt5-1)/2) and the author's exact interpolation coefficients in ℚ(sqrt5), we prove G₄(x₁,x₂) ≥ frac1500(3E²-1)(tfrac(3-sqrt5)2-E²) for 29/50 ≤ E ≤ 309/500, which is 93.40 band, an attained interior zero of G₄ on a threshold surface that the interpolation system forces but does not impose; these zeros are what limits every subdivision-based certificate, and the device that removes the first of them — a fold onto x₁ ≥ x₂ combined with an exact one-parameter deformation of the algebraic threshold — is what closes the first band. All positivity claims rest on the nonnegativity of explicit finite integer Bernstein coefficient arrays; every statement below is machine-checked in Lean 4. The passage from these polynomial inequalities to the Mourre estimate for D itself is the source's functional calculus, quoted rather than reproved.

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