  The nonlinear response is investigated for a space-fractional quantum
mechanical system subject to a static electric field. Expressions for the
polarizability and hyperpolarizability are derived from the fractional
Schr\"{o}dinger equation in the particle-centric view for a three-level model
constrained by the generalized Thomas-Rieke-Kuhn sum rule matrix elements.
These expressions resemble those for a semi-relativistic system, where the
reduction of the maximum linear and nonlinear static response is attributed to
the functional dependence of the canonical position and momentum commutator. As
examples, a clipped quantum harmonic oscillator potential and slant well
potential are studied. The linear and first nonlinear response to the
perturbing field are shown to decrease as the space fractionality is moved
further below unity, which is caused by a suppression of the dipole transition
moments. These results illustrate the importance of dimensionality and the
order of the kinetic momentum operator which affect the strength of a system's
optical response.
