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We use the voltage biased tip of a scanning force microscope at a temperatureof 35\,mK to locally induce the fractional quantum Hall state of $\nu=1/3$ in asplit-gate defined constriction. Different tip positions allow us to vary thepotential landscape. From temperature dependence of the conductance plateau at$G=1/3 \times e^2/h$ we determine the activation energy of this local $\nu=1/3$state. We find that at a magnetic field of 6\,T the activation energy isbetween 153\,$\mu$eV and 194\,$\mu$eV independent of the shape of the confiningpotential, but about 50\% lower than for bulk samples.

Investigating Energy Scales of Fractional Quantum Hall States using Scanning Gate Microscopy