  The effective action on long strings, such as confining strings in pure
Yang-Mills theories, is well-approximated by the Nambu-Goto action, but this
action cannot be exact. The leading possible corrections to this action (in a
long string expansion in the static gauge), allowed by Lorentz invariance, were
recently identified, both for closed strings and for open strings. In this
paper we compute explicitly in a Hamiltonian formalism the leading corrections
to the lowest-lying Nambu-Goto energy levels in both cases, and verify that
they are consistent with the previously computed effective string partition
functions. For open strings of length R the leading correction is of order
1/R^4, for excited closed strings of length R in D>3 space-time dimensions it
is of order 1/R^5, while for the ground state of the closed string in any
dimension it is of order 1/R^7. We attempt to match our closed string
corrections to lattice results, but the latter are still mostly outside the
range of convergence of the 1/R expansion that we use.
