effectively out of reach. In this article we develop a new formula (in section 2) which avoids this matrix inversion. This allows us to handle the cases of three and four impurities without much trouble. The resulting formulas for the dilatation operator are quite lengthy (see Appendix A), but their spectrum is surprisingly simple.

[*Our results suggest that for the class of operators considered, the Hamiltonian defined by the dilatation operator is integrable - it is just a set of oscillators. This is an example of an integrable dilatation operator, obtained by summing both planar and non-planar diagrams.* ]{}

The operators we consider can be mapped to giant gravitons[@McGreevy:2000cw] in spacetime[@Balasubramanian:2001nh; @Corley:2001zk; @Berenstein:2004kk]. There is in fact already a known connection between the geometry of giant gravitons and harmonic oscillators[@Mikhailov:2000ya; @Beasley:2002xv; @Kinney:2005ej; @Biswas:2006tj; @Mandal:2006tk; @jurgis]. Our work differs from these results in at least two important ways. Firstly, we claim that the complete spectrum (not just the BPS spectrum!) has a connection to harmonic oscillators. Secondly, we have very good control over the set of operators we consider. Our operators are dual to a two giant system. Previous studies captured the full set of BPS states and consequently were not 