The Schrödinger operators
with potentials p(x) which do not necessarily converge to a constant at infinity will
be discussed. The potential p(x) = x1∕|x|, x = (x1,x2,…,xn) ∈RN, is an example.
The radiation condition associated with such Schrödinger operators is shown to
have the form ∇u−i(∇R)u = small at infinity, where R = R(x,λ) is a solution of
the eikonal equation |∇R|2= 1 − p(x)∕λ. This radiation condition is “nonspherical”
in the sense that ∇R is not proportional to the vector x= x∕|x| in general. The
limiting absorption principle will be obtained using a priori estimates for the
radiation condition.
