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We give the quantization and spectrum of an RNS supersymmetric open 2-brane described by a Polyakov-like action, the model is world-volume supersymmetric. We present the Hamiltonian of the system in terms of raising and lowering operators. We get a supersymmetric spectrum of excited states in a discrete form after a GSO-like projection, which may be useful for further exploration related to the continuous spectrum of supermembranes.