Exact Sum Rules and Zeta Generating Formulas from the ODE/IM correspondence

TL;DR

The paper develops a spectral-zeta framework for PT-symmetric and Hermitian one-dimensional QMs tied to the ${ m A}_{2M-1}$ fusion hierarchy. By translating the ODE/IM fusion relations into exact sum rules and zeta generating formulas, it reveals how spectral data across PT and Hermitian sectors are redistributed and constrained, with an algebraic information-loss phenomenon for odd $M$. The approach shows that ESRs reproduce known exact-WKB results for Hermitian cases and systematically extend them to PT sectors, while ZGFs provide fixed-source, closed-form sector mappings and illuminate when these mappings are invertible. As an application, it recasts the massless Ai–Bender–Sarkar conjecture in purely spectral terms, linking PT and Hermitian spectra without analytic continuation. Overall, the work offers a structurally new, purely spectral pathway to understand non-Hermitian- Hermitian correspondences through the ODE/IM lens, with potential extensions to higher-rank integrable structures and resurgence phenomena.

Abstract

We develop a spectral-zeta framework for quantum mechanics with the ${\cal PT}$-symmetric potential $V_{\cal PT}(x)=x^{2K}(ix)^{\varepsilon}$ $(K,\varepsilon \in {\mathbb N})$ and the Hermitian potential $V_{\cal H}(x)=x^{2M}$ $(M \in {\mathbb N}+1)$, based on the fusion relations of the $A_{2M-1}$ T-system. Using the ODE/IM correspondence, we construct exact sum rules (ESRs) and zeta generating formulas (ZGFs) for the spectral zeta functions (SZFs) $ζ_n(s)$. In contrast to recursive T-Q relations, the ZGFs provide fixed-source, closed-form mappings between different fusion sectors. For Hermitian $M=2$, our ESRs reproduce exact WKB results, extending them systematically to ${\cal PT}$ sectors and (half-)integer $M$. Our analysis reveals a phenomenon of \textit{algebraic information loss}, distinct from analytic ambiguity. The structure is governed by a selection rule ${\cal S}_n$, derived from the Chebyshev structure of fusion relations and $\mathbb{Z}_{2M+2}$ Symanzik symmetry. For odd integer $M$, we identify a structural non-invertibility: mapping from \textit{odd} to \textit{even} fusion sectors causes exact coefficient cancellation due to phase interference, rendering the map non-invertible. This implies even-sector data carry strictly less information than odd-sector data, yielding a \textit{no-go} statement for inverse spectral reconstruction. Conversely, for even and half-integer $M$, all relevant sectors form an information-equivalent, mutually invertible family. Finally, we provide a spectral-zeta formulation of the massless Ai-Bender-Sarkar (ABS) conjecture. By connecting ${\cal PT}$ and Hermitian spectra via ZGFs, we establish a purely spectral-theoretic route to the conjectured relation, avoiding explicit analytic continuation.

Figures (5)

• Figure 1: The structure of the complex $x$-plane of QMs defined by the ${\cal PT}$-symmetric and the Hermitian-type operators, $\widehat{\cal L}_{\cal PT}(x,E)$ and $\widehat{\cal L}(x,E)$. The gray arrow and the black lines denote the real axis and the anti-Stokes lines, respectively. The ${\cal PT}$-symmetric QCs are given by analytic continuations represented by the blue and red arrows. The green arrow denotes the path for the Hermitian QC.
• Figure 2: The schematic figure of our strategy to formulate the ESRs and the ZGFs by beginning with the fusion relations. In this figure, the ESRs and the ZGFs are symbolically denoted by $\zeta_n(s) = \zeta_n(\{ \zeta_n(s^\prime ) \}_{1 \le s^\prime < s})$ and $\bm{\zeta}_n = \bm{\zeta}_n(\bm{\zeta}_{n^\prime})$. The detail is explained in the text.
• Figure 3: The schematic figures of $\Sigma_{k}^{(n)}$ for $n=5$ (Left) and $6$ (Right). The black circled numbers are elements of the sets in the family, $\Sigma_{k}^{(n)}$. $\Sigma^{(n)}_{\lfloor \frac{n+1}{2} \rfloor}$ only contains ${\bf A}_n$, and the sets which belong to $\Sigma^{(n)}_k$ can be recursively determined by subtracting the pair, $\{\widehat{a}_j,\widehat{a}_j + 2\}$, from those in $\Sigma^{(n)}_{k+1}$.
• Figure 4: The schematic figures of ${\Sigma}_{k}^{(M+1)}|_{\bmod (2 M + 2)}$ for $M=4$ (Left) and $5$ (Right). The black circled numbers are elements of the sets in the family, ${\Sigma}_{k}^{(M+1)}|_{\bmod (2 M + 2)}$. ${\Sigma}^{(M + 1)}_{k}|_{\bmod (2 M + 2)}$ is defined from $\Sigma^{(n)}_k$ in Eq.\ref{['eq:def_SigmaM']} by imposing the modulo identification, i.e., $\bmod (2 M + 2)$.
• Figure 5: The construction of $\Sigma^{(2M)}_{k}\mid_{\bmod (2M+2)}$ from $\Sigma^{(2M)}_{k}$ for $M=\frac{5}{2}$. After finding $\Sigma^{(2M)}_{k}$ for all $k$, imposing the ${\mathbb Z}_{2M+2}$ modulo identification to elements of the sets in the family yields $\Sigma^{(2M)}_{k}\mid_{\bmod (2M+2)}$. In the $M = \frac{5}{2}$ case, the elements, $\{\mp 4 \}$, in the sets are identified as $\{ \mp 4 \} \sim \{\pm 3\}$. The sets in the family $\Sigma_k^{(2M)} \mid_{2M+2}$ are in general subsets of ${\cal M} \setminus \{ \pm 1\}$. See also the left panel of Fig. \ref{['fig:Sigma']} for comparison.

Theorems & Definitions (2)

• Proposition 1: Structural non-invertibility
• proof