Positive Hamiltonians can give purely exponential decay

It is commonly claimed that only Hamiltonians with a spectrum unbounded both above and below can give purely exponential decay. Because such Hamiltonians have no ground state, they are considered unphysical. Here we show that Hamiltonians which are bounded below can give purely exponential decay. This is possible when, instead of looking at the global survival probability, one considers a subsystem only. We conclude that purely exponential decay might not be as unphysical as previously thought.