Perturbation Expansion for Distributions on Surfaces, Lagrange-Bürmann Theorem and Applications to Mechanics

Abstract

Let H ₀ be the Hamiltonian of an unperturbed oscillating system and let V be a small perturbation which we take to be analytic in a neighbourhood of the surface H ₀ = E > 0. If H = H ₀ + λV is the total Hamiltonian than there is a convergent perturbation expansion near λ = 0 of the form

$$\delta (H - E) = \sum\limits_{h = 0}^\infty {\frac{{{\lambda ^n}}}{{n!}}{V^n}{\delta ^n}({H_0} - E)}$$

((1))

when applied to test function which are analytic near H ₀ = E. In (1) δ (H - E) is the Dirac¹ distribution concentrated on the surface H = E and δ ⁽ⁿ⁾ H ₀ - E) are the derivatives of the Dirac distribution on the unperturbed surface H ₀ = E.