We have performed a numerical investigation of the ground state properties of the frustrated quantum Heisenberg antiferromagnet on the square lattice ("J_1-J_2 model"), using exact diagonalization of finite clusters with 16, 20, 32, and 36 sites. Using a finite-size scaling analysis we obtain results for a number of physical properties: magnetic order parameters, ground state energy, and magnetic susceptibility (at q=0). For the unfrustrated case these results agree with series expansions and

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... antum Monte Carlo calculations to within a percent or better. In order to assess the reliability of our calculations, we also investigate regions of parameter space with well-established magnetic order, in particular the non-frustrated case J_2<0. We find that in many cases, in particular for the intermediate region 0.3 < J_2/J_1 < 0.7, the 16 site cluster shows anomalous finite size effects. Omitting this cluster from the analysis, our principal result is that there is Néel type order for J_2/J_1 < 0.34 and collinear magnetic order (wavevector Q=(0,π)) for J_2/J_1 > 0.68. There thus is a region in parameter space without any form of magnetic order. Including the 16 site cluster, or analyzing the independently calculated magnetic susceptibility we arrive at the same conclusion, but with modified values for the range of existence of the nonmagnetic region. We also find numerical values for the spin-wave velocity and the spin stiffness. The spin-wave velocity remains finite at the magnetic-nonmagnetic transition, as expected from the nonlinear sigma