标题： 从Pöschl-Teller势和双曲几何中的二次波动性 显示英文标题

标题： Quadratic Volatility from the Pöschl-Teller Potential and Hyperbolic Geometry

摘要： 这项研究建立了在二次正态波动率（QNV）规范下广义Black-Scholes方程与双曲Pöschl-Teller势的稳态薛定谔方程之间的形式等价性。 一系列规范变换将金融定价算子映射到量子哈密顿量，揭示了波动率微笑作为双曲流形上扩散的直接表现，其几何结构由QNV多项式的判别式分类。 我们对金融哈密顿量进行了完整的谱分析，推导出其离散和连续谱，并从得到的本征函数构造了定价核，这些本征函数由经典特殊函数给出。 这个基于规范理论视角的分析框架，为衍生品定价提供了非平凡的基准，并提供了对市场异常现象的基本几何解释。 指出了通向可积系统和形式场论类比的未来研究方向。 显示英文摘要

摘要： This investigation establishes a formal equivalence between the generalized Black-Scholes equation under a Quadratic Normal Volatility (QNV) specification and the stationary Schr\"odinger equation for a hyperbolic P\"oschl-Teller potential. A sequence of canonical transformations maps the financial pricing operator to a quantum Hamiltonian, revealing the volatility smile as a direct manifestation of diffusion on a hyperbolic manifold whose geometry is classified by the discriminant of the QNV polynomial. We perform a complete spectral analysis of the financial Hamiltonian, deriving its discrete and continuous spectra and constructing the pricing kernel from the resulting eigenfunctions, which are given by classical special functions. This analytical framework, grounded in a gauge-theoretic perspective, furnishes a non-trivial benchmark for derivative pricing and provides a fundamental geometric interpretation of market anomalies. Future research trajectories toward integrable systems and formal field-theoretic analogies are identified.