By Per-Olov Lowdin
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Extra resources for Advances in quantum chemistry. / Volume 4
So P is permutable with H + and H - . 54) H(Z - P ) = H + . 57) T(Z- P ) = H’. Thus H - and H + are positive operators. Let U be a bounded operator permutable with H. Then U t , the adjoint of U, is also permutable with Hand, as a consequence of Wecken’s lemma, U and U t are permutable with T as well as with H + and H - . Permutability with H + implies that H +UP = UH’P = 0, therefore UPf is always in M, hence UP = PUP. 58) which completes the proof. The reduction can be applied not only to the operator H , but also to the equally self-adjoint operator H A= H - AZ, where 1 is any real number.
The following propositions may then be verified : (a) P(A) is a projection operator for any real 1. (b) P(1) - P ( p ) is a projection operator when 12 p. (c) The norm IIP(1)fll is a monotone increasing function of 1for any fixed f E 2,. Such projection operators play an important role in the theory of selfadjoint operators. 111. Spectral Theory of Bounded Self-Adjoint Operators A. Initial Formulation of the Spectral Problem The discussion of bounded operators is considerably simpler than that of the unbounded ones because the former can be readily extended so that they are defined over the entire Hilbert space.
Hence Hf = BHh, Hg = CHh. 26) demonstrate the asserted permutability of H with B and C. The permutability of B with C follows from CB = HBB = BHB = BC. 21) may be written in the form HCh = h - Bh, valid for every h in 8. Therefore HC=I-B. Now multiply C = HB by C. Then c2= CHB= HCB= Hence B = B2 ( I - B)B = B - B?. + C2. 27) The operator B is seen to be positive. 15) it follows f = Bh = 0, therefore, B is positive definite, and, moreover (IV. 16) ensures that the bound of B does not exceed 1. Hence 0 S (Bh, h) 5 (h, h) holds for every h in Sj.
Advances in quantum chemistry. / Volume 4 by Per-Olov Lowdin