|
Consider a path-integral
E x e ò 0 t V ( X )( s )) ds j ( X ( t ))
which is the solution to a diffusion version of the generalized Schro¨dinger's equation
\frac ¶ u ¶ t = Hu , u (0, x ) = j ( x )
. Here
H = A + V
, where A is an infinitesimal generator of a strongly continuous Markov Semigroup corresponding to the diffusion process
{ X ( s ),0 \leqslant s \leqslant t , X (0) = x }
. For
A = \frac12 D
and V replaced by
- V
one obtains
H
= - H = - \frac12 D + V
, which represents a quantum mechanical Hamiltonian corresponding to a particle of mass 1 (in atomic units) subject to interaction with potential V. This paper is concerned with computer calculations of the second eigenvalue of
- \frac12 D - \frac1
Ö
x 2 + y 2 + z 2
by generating a large number of trajectories of an ergodic diffusion process.
|
