Theoretical analysis demonstrates full-space Born-diagonal tracking for moving flea profiles in double wells, indicating robust quantum state control under physical concentration.
We study a fixed smooth nonnegative strict right-barrier flea in the one-dimensional semiclassical quartic double well when the scalar-amplitude profile varies with the semiclassical parameter. A preceding full-Hilbert analysis gives exact dressed C^1 control for every vanishing amplitude width but only for source profiles fixed independently of h, while a separate exact two-state analysis treats moving location mixtures g_h=k*nu_h. We bridge these two surfaces. The new mechanism is an adaptive physical-amplitude rescaling: for a location center a, the effective width is sh,a=w_h max(1,|a|) and the coefficient is renormalized into a fixed compact window. Physical concentration w_h Y_h -> 0 in probability makes these adaptive widths uniformly small on a probability bulk. An exact cancellation in the dressed spectral phase then prevents translation escape from amplifying phase derivatives. Combining bounded-center stationary-region control, large-center exact push-forward, double-tail truncation, and static graph return yields full-space Born-diagonal tracking for one fixed normalized k in L^1(R) and arbitrary physically concentrating Borel location laws nu_h. No smoothness, moment, compact-support, or additional time-error coupling is imposed on k. The two-coordinate universal criterion is inherited rather than claimed as a new law.
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Panasenko (2026) studied this question.
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