Brownian oscillator with time-dependent strength: a delta function protocol

We consider a classical Brownian oscillator of mass m driven from an arbitrary initial state by varying the stiffness $k(t)$ of the harmonic potential according to the protocol k(t)=k₀+a\,δ(t), involving the Dirac delta function. The microscopic work performed on the oscillator is shown to be W=(a²/2m)\,q²-a q v, where q and v are the coordinate and velocity in the initial state. If the initial distribution of q and v is the equilibrium one with temperature T, the average work is W =a²T/(2m\,k₀) and the distribution $f(W)$ has the form of the product of exponential and modified Bessel functions. The distribution is asymmetric and diverges as W→ 0. The system's response for $t>0$ is evaluated for specific models.