Optimal upper constants in essential-spectrum gap bounds and off-diagonal independence

Determine the sharp best-possible values of the universal upper constants γ+ and \tilde{γ}+ that appear in two inequalities for two-dimensional canonical systems: (i) the essential-spectrum gap bound R_H ≤ γ+/√α, where R_H is the radius of the gap around 0 in the essential spectrum of the operator part of the model relation A_H and α := limsup_{t→b}[(∫_t^b ξ_φ^T H(s) ξ_φ ds)·(∫_a^t ξ_{φ+π/2}^T H(s) ξ_{φ+π/2} ds)], and (ii) the off-diagonal independence bound R_H ≤ \tilde{γ}+·R_{H_d}, where H_d(t) is the diagonal Hamiltonian built from the directional entries ξ_φ^T H(t) ξ_φ and ξ_{φ+π/2}^T H(t) ξ_{φ+π/2}. Establish the minimal constants γ+ and \tilde{γ}+ (independent of H) for which these bounds hold for all canonical systems H satisfying the stated integrability condition.

Background

The discreteness criterion provides two-sided bounds for the radius R_H of the gap in the essential spectrum near zero, in terms of a quantity α that combines integrals of the directional diagonal entries of the Hamiltonian. A companion independence theorem compares R_H to the same quantity computed for a diagonal truncation H_d. Numerical values for the constants have been proposed and the lower bounds are known to be sharp, but the optimal upper constants have not been identified.

Resolving this question would pin down the exact quantitative form of the general discreteness criterion and the off-diagonal independence estimate, sharpening the quantitative part of these results and informing subsequent applications across canonical systems and related models.