Transcendental Trace Formulas For Finite-Gap Potentials

We show that formulas differing from classical analogues of rational trace formulas for algebraic-geometric potentials occur in the theory of finite-gap integration of spectral equations. The new formulas contain transcendental modular functions and hypergeometric series. They result in transcendental relations for theta functions.

💡 Research Summary

The paper investigates trace formulas within the finite‑gap integration framework, revealing that the classical rational trace expressions—derived from the algebraic‑geometric data of the spectral curve—are not the only possible representations. Starting from the standard setting, a second‑order linear spectral operator (L\psi=\lambda\psi) with a finite‑gap spectrum is associated with a hyperelliptic (or, in low genus, elliptic) Riemann surface (\Gamma). The traditional approach expresses traces of powers of (L) as rational symmetric functions of the branch points ({E_i}) of (\Gamma). While these rational formulas capture the algebraic structure, they ignore the deeper analytic features of the curve, especially its modular properties.

To address this limitation, the author introduces two transcendental ingredients: (i) modular functions (the Klein (j)‑invariant, theta constants (\vartheta_k(\tau)), etc.) built from the period matrix of (\Gamma), and (ii) hypergeometric series ({}_2F_1(a,b;c;z)) that naturally arise when expanding the Baker‑Akhiezer function in the spectral parameter. By normalizing the Baker‑Akhiezer function and expressing its logarithmic derivative in terms of theta functions, the author obtains an expansion whose coefficients are not rational but involve combinations of (\tau)‑dependent modular forms and hypergeometric terms.

The main result is a family of “transcendental trace formulas” \