Abstract
The eigenstate thermalization hypothesis (ETH), which dictates that all
diagonal matrix elements within a small energy shell be almost equal, is a
major candidate to explain thermalization in isolated quantum systems.
According to the typicality argument, the maximum variations of such matrix
elements should decrease exponentially with increasing the size of the system,
which implies the ETH. We show, however, that the typicality argument does not
apply to most few-body observables for few-body Hamiltonians when the width of
the energy shell decreases at most polynomially with increasing the size of the
system.
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