Krylov complexity and Wightman power spectrum with positive chemical potential in Schrödinger field theory

A bstract We study Krylov complexity in Schrödinger field theory in the grand canonical ensemble with chemical potential μ , with an emphasis on the qualitatively new features that arise for μ > 0. In this regime the fermionic Wightman power spectrum is effectively single-sided and sharply truncated at ω = μ , which induces a crossover in the Lanczos coefficients and signals a dynamical transition from a bulk-dominated regime to a spectral-edge-dominated regime: b n displays a two-stage linear growth (from an early-time slope π/β to an asymptotic slope 2 /β ), while a n bends from near-zero values to a linear descent with slope −4 /β . We provide analytic support for the resulting complexity growth from three complementary viewpoints: (i) using an SL(2 , ℝ) algebraic construction matched to the asymptotic Lanczos data, we show that the late-time Krylov complexity must grow quadratically, K ( t ) ∝ t 2 ; (ii) by analyzing engineered Wightman spectra with controlled decay and truncation, we identify single-sided exponential decay as the key spectral feature responsible for the quadratic asymptotics, while an approximately even two-sided exponential spectrum explains the early-time K ( t ) ~ sinh 2 ( πt/β ) behavior at large μ ; (iii) we formulate the problem in terms of orthogonal polynomials and estimate the crossover scale separating the early- and late-stage regimes. Overall, our results help clarify the role of chemical potential and spectral truncation in shaping operator growth and Krylov complexity in this non-relativistic quantum field theory setting.