  Considerable progress has recently been made with geometrical approaches to
understanding and controlling small out-of-equilibrium systems, but a
mathematically rigorous foundation for these methods has been lacking. Towards
this end, we develop a perturbative solution to the Fokker-Planck equation for
one-dimensional driven Brownian motion in the overdamped limit enabled by the
spectral properties of the corresponding single-particle Schr\"odinger
operator. The perturbation theory is in powers of the inverse characteristic
timescale of variation of the fastest varying control parameter, measured in
units of the system timescale, which is set by the smallest eigenvalue of the
corresponding Schr\"odinger operator. It applies to any Brownian system for
which the Schr\"odinger operator has a confining potential. We use the theory
to rigorously derive an exact formula for a Riemannian "thermodynamic" metric
in the space of control parameters of the system. We show that up to
second-order terms in the perturbation theory, optimal dissipation-minimizing
driving protocols minimize the length defined by this metric. We also show that
a previously proposed metric is calculable from our exact formula with
corrections that are exponentially suppressed in a characteristic length scale.
We illustrate our formula using the two-dimensional example of a harmonic
oscillator with time-dependent spring constant in a time-dependent electric
field. Lastly, we demonstrate that the Riemannian geometric structure of the
optimal control problem is emergent; it derives from the form of the
perturbative expansion for the probability density and persists to all orders
of the expansion.
