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We consider the properties of an observable (such as a single spin componentthat squares to the identity) when expressed as a matrix in the basis of energyeigenstates, and then truncated to a microcanonical slice of energies ofvarying width. For a quantum chaotic system, we model the unitary or orthogonalmatrix that relates the spin basis to the energy basis as a random matrixselected from the appropriate Haar measure. We find that the spectrum ofeigenvalues is given by a centered Jacobi distribution that approaches theWigner semicircle of a random hermitian matrix for small slices. For slicesthat contain more than half the states, there is a set of eigenvalues ofexactly $\pm 1$. The transition to this qualitatively different behavior athalf size is similar to that seen in other quantities such as entanglemententropy. Our results serve as a benchmark model for numerical calculations inrealistic physical systems.

Microcanonical Truncations of Observables in Quantum Chaotic Systems