A new, non-perturbative, coordinate space method is formulated to calculate the full and partial wave amplitudes for the potentialae^−λr/r withλ=+0. The basic ingredients are a plausible use of the point Coulomb wave function up to moderate distances and a Wronskian identity to take care of the large distance behaviour of integrands.

We consider the survival amplitudeA(t) for a normalized decaying state whose energy spectral density vanishes asymptotically as an inverse power. By using simple calculus a Taylor expansion ofA(t) is derived aroundt=0, the form of the remainder term identified, and a physical significance given to the other coefficients. It is shown that the Taylor remainder may contain logarithms oft besides powers.

The variational conditions implied by the most probable equilibrium distribution for a dilute gas are set up exactly in terms of the digamma function without necessarily invoking a Stirling approximation. Through a sequence of lemmas it is proved that, at any given kinetic temperature, there are three classes of self-consistent solutions characterized by the parameterβ$$\beta \bar \gtrless 0$$ 0 and by non-Maxwellian tails. These ambiguities persist even for a free ideal gas.

We address the problem of classical frictional motion under a potentialV possessing a barrier, apart from other possible confining and nonstationary terms. It is pointed out that the Green’s solution of the exact equation of motion can be reduced (under suitable conditions) either to an improved Rayleigh form or a non-Rayleigh form, the latter being outside the scope of the standard large-friction treatment of the Fokker-Planck equation. The resulting dissipationless dynamics involves an appropriately scaled potential which may have promising applications to quantum stochastic phenomena. Genuine dissipative corrections in regions far away from the barrier can be accounted for by the higher-order terms in our asymptotic expansions.