  We answer the question: If a vacuum sector Hamiltonian is regularized by an
energy cutoff, how is the one-kink sector Hamiltonian regularized? We find that
it is not regularized by an energy cutoff, indeed normal modes of all energies
are present in the kink Hamiltonian, but rather the decomposition of the field
into normal mode operators yields coefficients which lie on a constrained
surface that forces them to become small for energies above the cutoff. This
explains the old observation that an energy cutoff of the kink Hamiltonian
leads to an incorrect one-loop kink mass. To arrive at our conclusion, we
impose that the regularized kink sector Hamiltonian is unitarily equivalent to
the regularized vacuum sector Hamiltonian. This condition implies that the two
regularized Hamiltonians have the same spectrum and so guarantees that the kink
Hamiltonian yields the correct kink mass.
