Liouville Resonances, Følner Criterion, and Tensor Inheritance for Finite-Support Quasi-Periodic Schrödinger Operators

The paper studies Schrödinger operators H_ω = −Δ + V_ω on L²(ℝ^d) with d ≥ 2, where V_ω is a real quasi-periodic potential with finite Fourier support S ⊂ ℤ^l. The basic frequency vector ω = (ω_1, ..., ω_l) ∈ (ℝ^d)^l is assumed rationally independent: if n_1 ω_1 + ... + n_l ω_l = 0 with n_j ∈ ℤ, then n_1 = ... = n_l = 0. The potential has the Fourier expansion V_ω(x) = ∑_{n ∈ ℤ^l} V̂(n) exp(i(n·ω)·x), with V̂(−n) equal to the complex conjugate of V̂(n) and supp V̂ ⊆ S for a finite symmetric set S ⊂ ℤ^l \ {0}. One writes Q = |S|, M = max_{s ∈ S} ‖s‖, and ε = ‖V‖_1 = ∑_{s ∈ S} |V̂(s)|. For r ≥ 1 one defines the r-fold sumset S_r = S + ... + S (r times) and S_r* = S_r \ {0}; the scalar resonance minimum on the active scale is μ_r^(1)(ω) = min_{s ∈ S_r*} ‖s·ω‖, with the convention μ_r^(1)(ω) = ∞ when S_r* is empty; the active lattice is Λ(S) = ℤ[S] ⊆ ℤ^l. The fiber, or dual, operator acts on ℓ²(ℤ^l) by (H(k)u)_n = ‖k + n·ω‖² u_n + ∑_{s ∈ S} V̂(s) u_{n−s}. The diagonal part is self-adjoint on its natural domain and the potential part is bounded with norm at most ε by Young's inequality. For ρ > 0, δ > 0 and v ∈ ℝ^d \ {0} one defines the shifted spherical resonance strip B_v(ρ, δ) = { k ∈ ℝ^d : ‖k‖ = ρ, |2k·v + ‖v‖²| < δ }. This is the correct geometric object because the actual denominator in any resolvent-based estimate is ‖k + v‖² − ρ² = 2k·v + ‖v‖². One denotes by σ_{d−1} the normalized surface measure on the sphere ‖k‖ = ρ, so that σ_{d−1}(S^{d−1}(ρ)) = 1. For R ≥ 1 the conservative resonance bad set is Bad_cons(ρ, ε, R) = ⋃_{r=1}^{R} ⋃_{s ∈ S_r*} B_{s·ω}(ρ, ε^r/2); for a single order r one writes Bad_r(ρ, ε) = ⋃_{s ∈ S_r*} B_{s·ω}(ρ, ε^r/2). The logarithmic resonance cutoff used by polynomial multiscale schemes is R(ρ) = ⌈2 log ρ / log(1/ε)⌉. The adiabatic regime is S_r^ad(ρ) = { s ∈ S_r* : ‖s·ω‖ < ρ^(−1/2) }, the effective regime is S_r^eff(ρ) = S_r* \ S_r^ad(ρ); the effective bad set is Bad^eff(ρ, ε, R) = ⋃_{r=1}^{R} ⋃_{s ∈ S_r^eff(ρ)} B_{s·ω}(ρ, ε^r/2); the adiabatic bad set is Bad^adiabatic(ρ, ε) = Bad_cons(ρ, ε, R(ρ)) \ Bad^eff(ρ, ε, R(ρ)), and one has the decomposition Bad_cons(ρ, ε, R(ρ)) = Bad^eff(ρ, ε, R(ρ)) ∪ Bad^adiabatic(ρ, ε). A finite-volume restriction is defined as follows: for a finite box K ⊂ ℤ^l, one denotes by ℓ²(K) the subspace of ℓ²(ℤ^l) of functions supported on K, identified with ℂ^K; the finite-volume restriction H_ω(K; k) of the fiber operator H_ω(k) to K with Dirichlet boundary conditions is the bounded self-adjoint operator on ℓ²(K) given by (H_ω(K; k)u)_n = ‖k + n·ω‖² u_n + ∑_{s ∈ S : n − s ∈ K} V̂(s) u_{n−s}, for n ∈ K, with u extended by zero outside K. A pair (ω, V) admits a polynomial multiscale good sequence with constants (C_0, E_*) if for every E ≥ E_*, with ρ = √E, one can cover the sphere S^{d−1}(ρ) by angular patches and assign to each patch a finite-volume restriction H_ω(K; k) such that there exists an exceptional angular set Exc_E ⊂ S^{d−1}(ρ) satisfying three conditions: σ_{d−1}(Exc_E) ≤ E^{−c} for some c > 0 independent of E; Exc_E contains the conservative resonance bad set, Bad_cons(ρ, ε, R(ρ)) ⊆ Exc_E; and outside Exc_E the resolvent satisfies ‖(H_ω(K; k) − E)^(−1)‖ ≤ E^{C_0}. The second condition is essential: without it a definition that only asks for some small exceptional set would not interact correctly with the resonance obstruction. The multiscale scheme must exclude the conservative resonance set before closing the induction. The geometric core of the paper is the sharp estimate for the measure of a shifted spherical strip. Let v ∈ ℝ^d \ {0}, ρ > 0, δ > 0. If ‖v‖ > 2ρ + δ/(2ρ), then B_v(ρ, δ) = ∅. Otherwise σ_{d−1}(B_v(ρ, δ)) ≤ C_d min{1, δ/(ρ‖v‖) + √δ/ρ}, where C_d depends only on d. The proof goes as follows. By rotational invariance assume v = ‖v‖ e_d and write k = ρ(√(1−t²) η, t) with t = k_d/ρ ∈ [−1, 1] and η ∈ S^{d−2}. Then k·v = ρ‖v‖ t and the defining condition becomes |2ρ‖v‖ t + ‖v‖²| < δ, equivalently |t + a| < h where a = ‖v‖/(2ρ) and h = δ/(2ρ‖v‖). If ‖v‖ > 2ρ + δ/(2ρ), then the minimum of |2k·v + ‖v‖²| over ‖k‖ = ρ equals ‖v‖(‖v‖ − 2ρ) > δ, so the strip is empty. Assume the strip is nonempty. The push-forward of σ_{d−1} under t = k_d/ρ has density comparable to (1−t²)^((d−3)/2) on [−1, 1]. If d ≥ 3 this density is bounded and the measure of an interval of length at most 2h is at most C_d h, so σ_{d−1}(B_v(ρ, δ)) ≤ C_d δ/(ρ‖v‖), which is stronger than the claimed bound. The remaining case is d = 2, where the density is comparable to (1−t²)^(−1/2). Let I = [−1, 1] ∩ (−a−h, −a+h), of length at most 2h. An elementary one-dimensional estimate gives ∫_I (1−t²)^(−1/2) dt ≤ C (|I| + √|I|) ≤ C (h + √h); indeed away from the endpoints t = ±1 the density is bounded by a constant depending on the distance to the endpoints, giving a linear contribution, while near an endpoint the integral of (1−t²)^(−1/2) over an interval of length ℓ is of order √ℓ. If a ≤ 1/2, either h ≥ 1/4 in which case the normalized measure is at most 1 ≤ C h, or h < 1/4 and I stays a positive distance from ±1, giving a linear bound C h; hence σ_1(B_v(ρ, δ)) ≤ C h = C δ/(ρ‖v‖). If a > 1/2, then ‖v‖ > ρ, and therefore √h = √(δ/(2ρ‖v‖)) ≤ √δ/ρ; consequently σ_1(B_v(ρ, δ)) ≤ C (h + √h) ≤ C (δ/(ρ‖v‖) + √δ/ρ). Combining the cases proves the lemma. The term √δ/ρ is unavoidable in dimension d = 2; it comes from tangency of the shifted strip to the sphere. This is the geometric reason why smallness conditions in finite-support multiscale estimates naturally involve √ε, not merely ε. The covering obstruction follows immediately: for g_1, ..., g_N ∈ ℝ^d \ {0} one has σ_{d−1}(⋃_{j=1}^N B_{g_j}(ρ, δ)) ≤ C_d ∑_{j=1}^N (δ/(ρ‖g_j‖) + √δ/ρ). In particular, if the right-hand side is strictly less than 1, the union cannot cover the whole sphere. Two scalar arithmetic conditions are used. The scalar weak Diophantine condition on Λ(S) = ℤ[S] requires the existence of C_0 > 0 and α ≥ 0 such that ‖s·ω‖ ≥ C_0 ‖s‖^(−α) for every nonzero s ∈ Λ(S). The scalar sub-exponential condition with rate c_1 > 0 requires μ_r^(1)(ω) ≥ exp(−c_1 r) for every r ≥ 1. Scalar weak Diophantine implies the sub-exponential condition on S_r: if s ∈ S_r then ‖s‖ ≤ rM, hence μ_r^(1)(ω) ≥ C_0 (rM)^(−α) ≥ exp(−c_1 r) for a suitable c_1 = c_1(α, M, C_0). The conservative bad-set decay theorem states that, for a potential with finite Fourier support S and ε = ‖V‖_1, if either (i) ω satisfies the scalar weak Diophantine condition on Λ(S) and Q√ε < 1/2, or (ii) ω satisfies the scalar sub-exponential condition with rate c_1 and Q√ε exp(c_1) < 1/2, then with R(ρ) = ⌈2 log ρ / log(1/ε)⌉ one has σ_{d−1}(Bad_cons(ρ, ε, R(ρ))) ≤ C/ρ = C E^(−1/2), where E = ρ² and C depends only on d, Q, M and the arithmetic constants. The proof is a direct summation using the strip lemma and the bound |S_r*| ≤ Q^r. For each s ∈ S_r*, Lemma 3.1 gives σ_{d−1}(B_{s·ω}(ρ, ε^r/2)) ≤ C_d (ε^r/(ρ μ_r^(1)(ω)) + ε^(r/2)/ρ). Summing over s ∈ S_r* gives σ_{d−1}(Bad_r(ρ, ε)) ≤ (C_d/ρ)(Q^r ε^r/μ_r^(1)(ω) + Q^r ε^(r/2)), and therefore σ_{d−1}(Bad_cons(ρ, ε, R)) ≤ (C_d/ρ) ∑_{r=1}^{R} (Q^r ε^r/μ_r^(1)(ω) + Q^r ε^(r/2)). Under (i), for s ∈ S_r one has ‖s‖ ≤ rM, hence μ_r^(1)(ω) ≥ C_0 (rM)^(−α), so Q^r ε^r/μ_r^(1)(ω) ≤ (M^α/C_0) r^α (Qε)^r. Since Q√ε < 1/2 and Q ≥ 2, we have ε < 1 and hence Qε < 1/2; thus ∑ r^α (Qε)^r < ∞, and also ∑ (Q√ε)^r < ∞, giving the claim. Under (ii) one has Q^r ε^r/μ_r^(1)(ω) ≤ (Qε e^{c_1})^r, and the hypothesis Q√ε e^{c_1} < 1/2 implies Qε e^{c_1} < 1/2; again both series converge. The effective bad-set decay theorem is sharper and does not use exterior small divisors. For a potential with finite Fourier support S and ε = ‖V‖_1, assuming Q√ε < 1/2, one has, with R(ρ) = ⌈2 log ρ / log(1/ε)⌉, σ_{d−1}(Bad^eff(ρ, ε, R(ρ))) ≤ C ρ^(−1/2) = C E^(−1/4), where C = C(d, Q) is independent of ω, β and ε. The proof is again a summation. If s ∈ S_r^eff(ρ), then ‖s·ω‖ ≥ ρ^(−1/2), and Lemma 3.1 gives σ_{d−1}(B_{s·ω}(ρ, ε^r/2)) ≤ C_d (ε^r ρ^(−1/2) + ε^(r/2) ρ^(−1)). Summing over at most Q^r elements and over r ≤ R(ρ), one gets σ_{d−1}(Bad^eff) ≤ C_d ρ^(−1/2) ∑_{r≥1} (Qε)^r + C_d ρ^(−1) ∑_{r≥1} (Q√ε)^r. Since Q√ε < 1/2 both geometric series are bounded, and for ρ ≥ 1 one has ρ^(−1) ≤ ρ^(−1/2); hence σ_{d−1}(Bad^eff) ≤ C ρ^(−1/2), which is C E^(−1/4) because E = ρ². The limitation of this result is the exclusion of the adiabatic regime ‖s·ω‖ < ρ^(−1/2), where the effective bad-set estimate gives no control; the full conservative bad set can still be large when β > 1, and this is precisely what the sharpness theorem exploits. The construction of Liouville frequencies with simultaneous super-polynomial small divisors is the content of the simultaneous Liouville approximation proposition. Let β > 1 and c > 0. There exist α, γ ∈ (0, 1) such that 1, α, γ are linearly independent over ℚ, and such that there exist integers q_j → ∞ and p_j, r_j ∈ ℤ with |α − p_j/q_j| ≤ exp(−c q_j^β) and |γ − r_j/q_j| ≤ exp(−c q_j^β). The proof proceeds by nested closed squares. Enumerate all rational affine lines in ℝ² as L_1, L_2, ..., where each L_n is a set of the form a_n x + b_n y + d_n = 0 with integers a_n, b_n, d_n not all zero. Construct nested closed squares Q_j = z_j + [−η_j, η_j]², z_j = (p_j/q_j, r_j/q_j), η_j = exp(−c q_j^β), such that Q_{j+1} ⊂ Q_j and Q_j ∩ L_n = ∅ for every 1 ≤ n ≤ j. Start by choosing a small open square U_1 compactly contained in (0, 1)² and disjoint from L_1; choose q_1 so large that the grid q_1^(−1) ℤ² meets U_1 and η_1 is smaller than the distance from U_1 to ∂(0, 1)² and to L_1; pick z_1 ∈ U_1 ∩ q_1^(−1) ℤ² and set Q_1 = z_1 + [−η_1, η_1]², so that Q_1 ⊂ U_1 and Q_1 ∩ L_1 = ∅. For the induction step, assume Q_j has been constructed. Since L_1, ..., L_{j+1} are finitely many lines, there exists an open ball U_{j+1} compactly contained in the interior of Q_j and disjoint from all of them; choose q_{j+1} so large that the grid q_{j+1}^(−1) ℤ² meets U_{j+1} and η_{j+1} = exp(−c q_{j+1}^β) is smaller than the distance from U_{j+1} to ∂Q_j and to e