Spectral Structure of the Mixed Hessian of the Dispersionless Toda $\tau$-Function

Abstract

We study the mixed Hessian of the dispersionless Toda $\tau$-function for the one-harmonic $s$-fold symmetric conformal map $f(w)=rw+aw^{1-s}$. This Hessian is the susceptibility matrix generated by the inverse conformal map. Our spectral statements are formulated for its weighted symmetry-block realizations on a fixed Hilbert space. In that realization, the first spectral transition occurs at the analytic threshold $\zeta_c=(s-1)^{s-1}/s^s$, where the dominant square-root singularity of the inverse map reaches the normalization circle, rather than at the geometric threshold $\zeta_{\mathrm{univ}}=1/(s-1)$, where univalence fails. After symmetry decomposition and weighted realization, each block develops exactly one logarithmically diverging eigenvalue as $\zeta\uparrow\zeta_c$, while the remaining spectrum stays bounded and converges to a compact limit. The instability is therefore rank one in every symmetry sector of the weighted block theory. We then continue the scalar Gram generating functions beyond $\zeta_c$. They are generalized hypergeometric functions on the slit plane $\mathbb{C}\setminus[\zeta_c^2,\infty)$, their branch-point expansion contains the logarithmic term responsible for the divergence, and in the range $1\le p\le s$ they admit Cauchy--Stieltjes and Jacobi-matrix realizations. In particular, the continued scalar quantities remain regular at $\zeta_{\mathrm{univ}}$, so the analytic spectral transition strictly precedes the geometric breakdown of univalence.