  We present preliminary numerical evidence for the hypothesis that the
Hamiltonian SU(2) gauge theory discretized on a lattice obeys the Eigenstate
Thermalization Hypothesis (ETH). To do so we study three approximations: (a) a
linear plaquette chain in a reduced Hilbert space limiting the electric field
basis to $j=0,\frac{1}{2}$ , (b) a two-dimensional honeycomb lattice with
periodic or closed boundary condition and the same Hilbert space constraint,
and (c) a chain of only three plaquettes but such a sufficiently large electric
field Hilbert space ($j \leq \frac{7}{2})$ that convergence of all energy
eigenvalues in the analyzed energy window is observed. While an unconstrained
Hilbert space is required to reach the continuum limit of SU(2) gauge theory,
numerical resource constraints do not permit us to realize this requirement for
all values of the coupling constant and large lattices. In each of the three
studied cases we check first for random matrix theory (RMT) behavior in the
eigenenergy spectrum and then analyze the diagonal as well as the off-diagonal
matrix elements between energy eigenstates for a few operators. Within current
uncertainties all results for (a), (b) and (c) agree with ETH predictions.
Furthermore, we find the off-diagonal matrix elements of the electric energy
operator exhibit RMT behavior in frequency windows that are small enough in (b)
and (c). To unambiguously establish ETH behavior and determine for which class
of operators it applies, an extension of our investigations is necessary.
