By Per-Olov Lowdin
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Extra resources for Advances in quantum chemistry. / Volume 4
48) + Then as n tends to co,we obtain H 2 = Z - 2Y Y 2 = (Z- Y)’, where T = Z - Y is a positive operator. 49) H 2 = T2. The permutability of T with every bounded operator permutable with H follows from the permutability of all Y‘s with such operators. D. Reduction of a Bounded Self-Adjoint Operator We shall prove the following theorem: Given a bounded, self-adjoint operator H there exists a projection operator P permutable with H and such that (a) When Hf = 0 then Pf =f, (b) HP is a negative operator; (HPf,f) 5 0, (c) H(Z - P ) is a positive operator, (H(Z - P ) f ,f ) 2 0.
B) Consider for the moment the multiplication operator in the space Q2 over a finite interval, say Q2[0,11. The elements of this space are the functions f ( x ) for which /011f(x)12dx is finite. The multiplication operator B, transforms f(x) into xf(x). But, since IIBFfI12= lo1 Ixf(x)12 dx 5 sd If(x)12 dx = Ilf1I2, the operator B, is bounded and defined over Q2[0, 11. Since it is clearly a symmetric operator, it is also self-adjoint. The multiplication operator B in the space Q2 over an infinite interval is no longer bounded.
Unfortunately, it involves Stieltjes integration according to a complex function; it can, however, be reduced to integrations according to bounded monotone-increasing, real functions. 3). These equations are applicable to any self-adjoint operator, in particular to the operator E d , therefore ( E , A g ) is calculable as the sum of four functions of the type (E,h, h). 19) is the sum of four integrals according to bounded, monotone-increasing functions. For any fixed f this integral is a bounded linear functional of g because (E,S,g) is such a functional and, Ei being a finite resolution of the identity, the integration is confined to a finite interval.
Advances in quantum chemistry. / Volume 4 by Per-Olov Lowdin