The Question
When a magnetic potential A with horocycle flux \alpha \notin 2\pi\mathbb{Z} is introduced on \text{PSL}(2,\mathbb{Z})\backslash\mathbb{H}, the Morame–Truc theorem (2011) guarantees that the essential spectrum of -\Delta_A collapses entirely — and with it, the continuous spectrum of Eisenstein series through which the nontrivial zeros of \zeta(s) are classically encoded in the Selberg zeta function.
Define the spectral zeta function of the magnetic bottle:
where eigenvalues are counted with multiplicity and the sum converges for \text{Re}(s) sufficiently large by the Weyl law of Morame–Truc.
The question: does Z_A(s) retain a Stokes line at \text{Re}(s) = 1/2, and if so, through what mechanism — given that the scattering channel \varphi(s) = \xi(2s-1)/\xi(2s) that links Z_\Gamma(s) to \zeta(s) no longer exists?
Background: Two Results That Do Not Yet Talk to Each Other
Result A — Morame–Truc (2011).
Consider -\Delta_A = (id+A)^*(id+A) on a noncompact hyperbolic surface of finite area. Define the horocycle flux at the cusp:
Then:
If \alpha \notin 2\pi\mathbb{Z}: \text{sp}_\text{ess}(-\Delta_A) = \emptyset. Spectrum is purely discrete, satisfying the Weyl law (magnetic bottle).
If \alpha \in 2\pi\mathbb{Z}: \text{sp}_\text{ess}(-\Delta_A) = [1/4 + b^2, +\infty).
Result B — Berry (1989, 1995).
On the critical line s = 1/2 + it, the Riemann–Siegel expansion of \zeta(s) is divergent. Terms decrease until r^* \approx 2\pi t then increase; the remainder upon optimal truncation is R(t) = O(e^{-\pi t}). The critical line is therefore a Stokes line for this expansion, with universal smooth switch-on:
The Standard Pathway and Its Obstruction
The connection between the spectral geometry of \text{PSL}(2,\mathbb{Z})\backslash\mathbb{H} and \zeta(s) is structural. Via the Selberg trace formula:
The zeros of \zeta(s) enter through the scattering matrix of the Eisenstein series:
If \zeta(s_0) = 0, then \varphi(s) has a pole at s_0, so Z_\Gamma(s) acquires a zero there. The nontrivial zeros of \zeta(s) are encoded in Z_\Gamma via the continuous spectrum of Eisenstein series.
A clarification on the Stokes transfer. Berry's result concerns the asymptotic expansion of \zeta(s) directly, not of Z_\Gamma(s). The claim that Z_\Gamma(s) inherits a Stokes line at \text{Re}(s) = 1/2 is not explicitly established in the literature. What is established is that the zeros of \zeta(s) appear as zeros of Z_\Gamma(s) via \varphi(s). Whether this zero-sharing implies sharing of resurgent structure in the Borel plane is part of what this question is asking. The question is therefore: does Z_A(s) — which loses the \varphi(s) channel entirely — retain any trace of the Stokes geometry of \zeta(s), and if so, is that trace visible already in Z_\Gamma(s) before the magnetic deformation?
The obstruction: when \alpha \notin 2\pi\mathbb{Z}, Morame–Truc eliminates the entire continuous spectrum. The scattering matrix \varphi_A(s) is no longer defined. The standard pathway closes.
The Dichotomy
If the Stokes line survives in Z_A(s): it must do so through the discrete spectrum alone, requiring a purely arithmetic or geometric mechanism independent of scattering.
If the Stokes line does not survive: the Stokes line at \text{Re}(s) = 1/2 is topologically linked to the existence of the Eisenstein scattering channel — a new geometric interpretation of the critical line.
Both outcomes are significant.
Two Focused Subquestions
Q1 (Analytic continuation). The Weyl asymptotics guaranteed by Morame–Truc ensure that Z_A(s) converges for \text{Re}(s) sufficiently large. Whether it admits meromorphic continuation to \mathbb{C} is plausible by analogy with the compact case (Finski 2022) but not established for the noncompact modular surface with cusp. Granting this continuation as a working hypothesis: what replaces the functional equation when \varphi_A(s) is absent? And is there a normalization such that Z_A(s) \to Z_\Gamma(s) as \alpha \to 0? Q2 below is posed under this same hypothesis.
Q2 (Stokes remainder at the threshold). As \alpha \to 2\pi\mathbb{Z} (the bottle collapses and the essential spectrum reappears), does the Stokes remainder of Z_A(s) blow up in a way that mirrors the divergence of the Riemann–Siegel series near its Stokes line? A concrete version: does the remainder satisfy
for some \beta > 0 as \alpha \to 2\pi\mathbb{Z}, and if so, what determines \beta?
The ansatz is motivated as follows. As \alpha \to 2\pi\mathbb{Z}, the essential spectrum re-enters at threshold 1/4 + b^2 by Morame–Truc, and the discrete eigenvalues accumulate toward this threshold. The exponential factor e^{-\pi t} is inherited from Berry's Stokes remainder for \zeta(s); the algebraic prefactor \text{dist}(\alpha, 2\pi\mathbb{Z})^{-\beta} reflects the expected divergence of the remainder as the scattering channel reopens. The exponent \beta is undetermined; finding it — or showing it does not exist — is the content of the question.
What Is Known and What Is Not
The individual pieces are established:
Magnetic Laplacians on hyperbolic cusps: Morame–Truc (2011), Golénia–Moroianu (2008), Lefeuvre–Charles–Chabert (arXiv:2601.04804, 2025).
Stokes structure of the Riemann–Siegel expansion: Berry (1989, 1995), Berry–Keating (1992).
Selberg–Riemann connection: Hejhal (1976, 1983), Fischer (1987).
Resurgence of WKB solutions on Riemann surfaces: Nikolaev (arXiv:2410.17224, 2024).
Selberg trace formula for compact magnetic surfaces: Finski (arXiv:2202.06055, 2022) — does not cover the noncompact modular surface with cusp.
The resurgent structure of Z_A(s) on the noncompact modular surface under the Morame–Truc flux condition does not appear in any of these works.
References
Berry, M.V. (1989). Proc. R. Soc. Lond. A422, 7–21.
Berry, M.V. (1995). Proc. R. Soc. Lond. A450, 439–462.
Berry, M.V. & Keating, J.P. (1992). J. Phys. A25, 4839–4847.
Finski, S. (2022). arXiv:2202.06055.
Fischer, J. (1987). LNM 1253, Springer.
Golénia, S. & Moroianu, S. (2008). Ann. Henri Poincaré 9, 131–177.
Hejhal, D.A. (1976, 1983). LNM 548 & 1001, Springer.
Lefeuvre, T., Charles, L. & Chabert, A. (2025). arXiv:2601.04804.
Morame, A. & Truc, F. (2011). Lett. Math. Phys. 97, 203–211.
Nikolaev, N. (2024). arXiv:2410.17224.
Selberg, A. (1956). J. Indian Math. Soc. 20, 47–87.
This question was formulated by Jean Terán. The central dichotomy and the conjecture in Q2 are the author's. Mathematical exposition prepared with the assistance of Claude (Anthropic). Dedicated to Dana Fernández.
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