Titchmarsh determined the
spectrum of the Schrödinger energy operator associated with the hydrogen
atom, i.e. the operator −Δ − 1∕r. He showed, in particular, that its essential
spectrum consists of the positive real axis. On the other hand, Agudo-Wolf
and Birman formulated overlapping criteria for a potential, which ensured
that addition of such a potential does not change the essential spectrum of
−Δ.
These criteria do not admit the potential 1∕r and a criterion admitting it is
formulated in the forthcoming work of Balslev where he also considers operators in
Lp spaces. In this paper we slightly extend this Balslev criterion, in case the operator
is a Schrödinger operator. Our proofs are different, inasmuch as we capitalize on the
representation of the kernel of the unperturbed resolvent. Then we make essential use
of a result of Friedrichs which gives a bound for the norm of an integral
operator.
