Analytic spread complexity for second Toda equations beyond special ratios

Derive analytical expressions for the spread complexity K(t, τ2) in the setting of the second Toda equations with vanishing diagonal Lanczos coefficients a_n = 0 and alternating off-diagonal coefficients b_{2n+1}^2(τ2) = (2n+1) γ^2(τ2) and b_{2n}^2(τ2) = 2n α^2(τ2), for generic values of the ratio γ/α not equal to 0, 1, or ∞.

Background

The paper analyzes deformations governed by generalized Toda flows in Krylov space and provides exact solutions for several cases. For the second Toda equations with a_n = 0, the authors introduce an alternating ansatz for the off-diagonal Lanczos coefficients that yields block-diagonal structure and closed-form solutions in special cases (γ/α = 0, 1, ∞).

However, beyond these special ratios, the authors report that they do not have analytical solutions for the spread complexity and present numerical results instead. Establishing closed-form expressions in the generic case would complete the analytic characterization of this family of deformations and clarify how complexity behaves across parameter regimes where closed complexity algebras are not readily available.