Showing papers on "Gaussian measure published in 1979"

The Semiclassical Expansion of the Anharmonic-Oscillator Propagator.

Abstract: This paper shows how to calculate the terms of a semiclassical (WKB) expansion of the quantum‐mechanical propagator corresponding to the quartic anharmonic‐oscillator potential, V=mω2q2/2 +λq4/4. This nonperturbative treatment expresses each term in the series as a path integral, which is then evaluated in the framework of a formalism, introduced by C. DeWitt‐Morette, which does not entail the usual time‐slicing operation followed by a limiting procedure. The Gaussian measure used absorbs all the quadratic terms in the expansion of the action functional about a classical path. The covariance of this Gaussian measure is Feynman’s Green function for the small‐disturbance operator of the system. This function can be obtained by varying the constants of integration in the classical solution, and therefore the coefficients of the expansion depend only on this classical solution. If the latter is chosen to be the one which tends to its harmonic counterpart when λ→0, then it is seen that the propagator also tend...

Hilbertian support of a probability measure on a banach space

Abstract: In this paper, we will prove the following : Let E be a real separable Banach space. Then every probability measure on E has a Hilbertian support if and only if E is isomorphic to a Hilbert space. In the case of lp (1 ≤ p < 2) we will give an explicit construction of probability measures without Hilbertian support.