  It is commonly believed that quantum isolated systems satisfying the
eigenstate thermalization hypothesis (ETH) are diffusive. We show that this
assumption is too restrictive, since there are systems that are asymptotically
in a thermal state, yet exhibit anomalous, subdiffusive thermalization. We show
that such systems satisfy a modified version of the ETH ansatz and derive a
general connection between the scaling of the variance of the offdiagonal
matrix elements of local operators, written in the eigenbasis of the
Hamiltonian, and the dynamical exponent. We find that for subdiffusively
thermalizing systems the variance scales more slowly with system size than
expected for diffusive systems. We corroborate our findings by numerically
studying the distribution of the coefficients of the eigenfunctions and the
offdiagonal matrix elements of local operators of the random field Heisenberg
chain, which has anomalous transport in its thermal phase. Surprisingly, this
system also has non-Gaussian distributions of the eigenfunctions, thus directly
violating Berry's conjecture.
