  The Efimov effect can be induced by means of an external deformed one-body
field that effectively reduces the allowed spatial dimensions to less than
three. To understand this new mechanism, conceptually and practically, we
employ a formulation using non-integer dimension, which is equivalent to the
strength of an external oscillator field. The effect most clearly appears when
the crucial two-body systems are unbound in three, but bound in two,
dimensions. We discuss energy variation, conditions for occurrence, and number
of Efimov states, as functions of the dimension. We use practical examples from
cold atom physics of $^{133}$Cs-$^{133}$Cs-$^{133}$Cs,
$^{87}$Rb-$^{87}$Rb-$^{87}$Rb, $^{133}$Cs-$^{133}$Cs-$^{6}$Li, and
$^{87}$Rb-$^{87}$Rb-$^{39}$K. Laboratory tests of the effect can be performed
with two independent parameters, i.e. the external one-body field and the
Feshbach two-body tuning. The scaling and (dis)appearance of these Efimov
states occur precisely as already found in three dimensions.
