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bibkey: fucci2024poschlteller authors: Guglielmo Fucci; Jonathan Stanfill year: 2024 title: “The exotic structure of the spectral ζ-function for the Schrödinger operator with Pöschl–Teller potential” doi: 10.1007/s00023-025-01587-7 url: https://arxiv.org/abs/2411.17860v1 claim: “Remark B.3 conjectures that all g_{m,j}(p/q,β) are nonzero for the considered coprime 0 < p < q and nondegenerate β; the coefficient g_{3,1}(2/3,π/4) vanishes.” strata_touched:

• D5/S3/Analytic/SpectralZeta/PoschlTellerCoefficientVanishing license: citation-only triage: anchor

Verified locator

DOI: https://doi.org/10.1007/s00023-025-01587-7

Source: https://arxiv.org/abs/2411.17860v1

Appendix B, Remark B.3, printed page 31 (continuing on page 32), states:

From calculations performed for particular choices of $p$, $q$, and $\beta$, we in fact conjecture that all of the $g_{m,j}(p/q,\beta )$ are nonzero for the choices of $p,q,$ and $\beta$ considered here.

The journal locator is Annales Henri Poincaré 27 (2026), 2073–2116. The journal text retains Remark B.3.

Coefficient definitions

Appendix B’s general definitions determine the coefficients; the special display of ${\mathcal{P}}_1$ is not used as a definition. The manuscript labels used here are (B.17), $C_m$, $G_k$, ${\mathcal{P}}_k$, ${\mathcal{E}}_j$, (B.24)–(B.26), and (B.39)–(B.45). The typeset arXiv PDF numbers the final coefficient expansion (B.48).

The generalized Bernoulli convention is DLMF 24.16.1:

$${\left(\frac{t}{e^t-1}\right)}^a{e}^{xt}=\sum _{n=0}^{\infty }{B}_n^{(a)}(x)\frac{t^n}{n!}.$$

For real $a$, the power is the binomial series with descending Pochhammer coefficients. Mathlib’s bernoulliPowerSeries supplies $t/(e^t-1)$. With $T$ representing the common logarithmic denominator factor, ${\mathcal{E}}_0=1$ and ${\mathcal{E}}_j=E_{j}T/2$ for $j\ge 1$.

$$\begin{array}{rl}{C}_m(y)& =\sum _{j=0}^{m-1}\left(\genfrac{}{}{0ex}{}{2m-1}{2j}\right)\frac{2^{2m}{y}^{2j}}{m-j}{B}_{2(m-j)},\\ E_k(y)& =2(2y{)}^{2k-1}-\frac{(2y{)}^{2k}}{k}-C_k(y),\\ G_k(x,y)& =\left(\genfrac{}{}{0ex}{}{x-y}{k}\right)B_k^{(x-y+1)}(x),\\ {\mathcal{P}}_0(y)& =1,{\mathcal{P}}_k(y)=\sum _{j=0}^k{4}^{k-j}{\mathcal{E}}_j(y)G_{2(k-j)}(1-y,y)(k\ge 1).\end{array}$$

The factor ${\Omega }_0$ is $2^{2\nu -1}\Gamma (1+\nu )\cot (\beta )e^{i\pi \nu }/\Gamma (-\nu )$. Writing $y_{\pm }=(1\pm \nu )/2$, (B.24)–(B.26) give ${\bar{\mathcal{P}}}_k={\Omega }_0{\mathcal{P}}_k(y_{+})$ and ${\Omega }_k={\bar{\mathcal{P}}}_k-{\sum }_{l=0}^{k-1}{\mathcal{P}}_{k-l}(y_{-}){\Omega }_l$. The coefficient of $x^{m+p}$ in $\log (1+{\sum }_{k\ge 0}{\Omega }_k{x}^{p+kq})$ is ${\mathcal{S}}_m$; the coefficient of $T^j$ in ${\mathcal{S}}_m$ is $g_{m,j}$. The admissible indices satisfy $lp+kq=m$ with $l,k\in {\mathbb{N}}_0$, and $k_m$ is the largest such $k$.

Printed page 31 states:

The order of ${\mathcal{S}}_m(p/q,z)$ is, then, $k_m=\text{max}\{k_i\}$ with $i\in \mathbb{N}$, namely the largest value of $k_i$ amongst the vectors $(l_i,k_i)\in [m{]}_{p,q}$.

Reflection cancellation and scope

The Lean module proves the counterexample $(p,q,\beta ,m,j)=(2,3,\pi /4,3,1)$ directly from these definitions. At this order the formal logarithm contributes only ${\Omega }_1$. Since $G_0=1$, the $T$-coefficient of ${\mathcal{P}}_1(y)$ is $E_1(y)/2$. The identity $E_1(1-y)=E_1(y)$ cancels that coefficient at $y_{+}$ and $y_{-}$. The compiled private calculation gives $g_{3,1}(2/3,\beta )=0$ for every real $\beta$; the public result refutes the source’s universal conjecture.

The family $g_{q,1}(p/q,\beta )=0$ for all coprime $0<p<q$ follows in prose from $(n-1)p+kq=q$. For $p\ge 2$ the only pair is $(n,k)=(1,1)$; for $p=1$ the additional term $(q+1,0)$ is independent of $T$. This all-parameter family is not a theorem in this Lean module. For $p\ge 2$, the prose calculation gives $g_{q,0}={\Omega }_0\nu ({\nu }^2-1)/3\ne 0$ in the source’s nondegenerate range. Consequently the source’s assumption that all these coefficients are nonzero has no instance, and its $j=1$ branch-point contribution at $m=q$ is absent. These analytic interpretations use the source’s expansion; spectral continuation is not formalized here.

Computed evidence and remaining questions

The experiment entry docs/reports/fucci-stanfill-2024-poschl-teller-coefficients/check.py has SHA-256 50fcf972778131ed652e02d4b8998daea3dcb0f7e17a7042835b3ddded112909. The supplied reading is python3 check.py, exit 0, final line ALL_OK. It checks every coprime $0<p<q\le 8$, with symbolic ${\Omega }_0$ and arbitrary higher coefficients. This is computed evidence, not a Lean proof of the family.

It also checks the discrepancy in (B.21): the general definitions give the $T$-coefficient $-(6y^2-6y+1)/3$, whereas the displayed expression prints $-(6y^2-6y-1)/3$. Both are reflection symmetric, so either gives the same cancellation. This discrepancy is a computed source correction.

Open: classify other vanishing $g_{m,j}$, especially top-order terms arising from one reflection-symmetric product. Open: determine whether restrictions such as $j\ge 2$ or $m\notin q\mathbb{N}$ give a corrected genericity hypothesis for Sections 3.1.1–3.1.2.