  Wave functions, phase shifts and corresponding elastic cross sections are
investigated for two short-range interacting particles in a deformed external
oscillator field. For this we use the equivalent $d$-method employing a
non-integer dimension $d$. Using a square-well potential, we derive analytic
expressions for scattering lengths and phase shifts. In particular, we consider
the dimension, $d_E$, for infinite scattering length, where the Efimov effect
occurs by addition of a third particle. We give explicitly the equivalent
continuum wave functions in $d$ and ordinary three dimensional (3D) space, and
show that the phase shifts are the same in both methods. Consequently the
$d$-method can be used to obtain low-energy two-body elastic cross sections in
an external field.
