Vol. 19, No. 1, 1966

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On the essential spectrum of the hydrogen energy and related operators

Peter Alexander Rejto

Vol. 19 (1966), No. 1, 109–140

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.

Mathematical Subject Classification
Primary: 35.80
Secondary: 81.35
Received: 22 March 1965
Revised: 29 September 1965
Published: 1 October 1966
Peter Alexander Rejto