Geometric diagnostics of scrambling-related sensitivity in a Bohmian preparation space

Out-of-time-ordered correlators (OTOCs) provide algebraic diagnostics of operator growth and scrambling-related sensitivity, while Lagrangian descriptors (LDs) provide trajectory-based geometric diagnostics of finite-time phase-space structure. This article develops a preparation-space LD construction for localized Gaussian quantum states and compares it with semiclassical stability structures entering OTOC calculations. Motivated by Bohmian mechanics, the numerical diagnostic uses wavepacket-center (WPC) trajectories from direct wavepacket propagation. Gaussian preparations are labeled by parameters (q0, p0) representing initial center and mean momentum. The LD is evaluated on the induced preparation-space map (q0,p0)↦(qc(t),pc(t))=(⟨q̂⟩t,⟨p̂⟩t), where ⟨⋅⟩t denotes expectation at time t. The inverted harmonic oscillator provides an exactly solvable benchmark: WPC dynamics coincide with classical hyperbolic flow, and the preparation-space Jacobian is the usual stability matrix. The nonquadratic double-well potential V(q) = q4 - 2q2 is treated by split-operator propagation. The classical LD resolves the separatrix skeleton sharply, whereas the quantum WPC LD produces a smoother, preparation-dependent sensitivity landscape organized relative to the classical separatrix. Finite-time sensitivity maps compare classical and WPC final-time maps, including the canonical derivative ∂qc(T)/∂q0 relevant to semiclassical OTOCs. These results support preparation-space LDs as geometric companion diagnostics for semiclassical OTOC sensitivity, clarifying that an LD is neither an OTOC nor a universal limit of one.