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A first-order perturbative theory of the motion of a harmonic oscillator interacting with a weak arbitrary force field is presented, as it pertains to dynamic force microscopy. In essence the theory corresponds to a Born approximation for the scattering of standing waves trapped in a perturbed parabolic potential. In particular, it is shown that the scattering amplitudes are related to corresponding moments, involving Chebyshev polynomials and associated metrics of the conservative interaction force, and of a generalized friction coefficient accounting for irreversible interactions. Implications for dynamic force microscopy are discussed.
U. Dürig (2000) studied this question.
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