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e"1 g((l + e)x) q{x) | g q0(x) G Qt(Rr) holds for 0 < e < e 0 and some/3>0; in particular we have rqr(x) èqo(x); hence rqr£.Qp(R ). Under these conditions we shall prove in §2 a very general form of the Virial Theorem of quantum mechanics. In §§3 and 4 this theorem will be used to deduce some results on the spectrum of H. Let L2(R ) be the Hubert space of functions which are squaresummable over R; the inner product in this space will be denoted by ( • , • ), the norm by | • | . From condition (I) one can conclude (e.g. Ikebe-Kato [2]): (1) The operator H with domain D(H)=H2(R ) is selfadjoint in L2(R ) (H2(R ) is the closure of Co(R) with respect to the norm k | 2 = { Z ; , * Idtyidxfixà^+^sldu/dxjlt+lul*}"*). (2) For uGD(H) and qGQa(R ) we have quEL2(R ). (3) For u, vED(H) we have Au, AvEL2(R ) and (Au, v) = (u, Av).

The virial theorem and its application to the spectral theory of Schrödinger operators