Abstract: The authors study a class of solutions,namely, regular solutions of the Schrödinger equation (1/2△+q)u=0 on unbounded domains. They definite the regular solutions in terms of sample path properties of Brownian motion and then characterize them by analytic method. In Section 4, they discuss the regular solution to the stochastic Dirichlet problem for the equation (1/2△+q)u=0 having limit α at infinity.

Key words: Schrödinger equation, Regular solution, Stochastic Dirichlet problem