# Download PDF by Per-Olov Lowdin: Advances in quantum chemistry. / Volume 4

By Per-Olov Lowdin

ISBN-10: 0120348047

ISBN-13: 9780120348046

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**Extra resources for Advances in quantum chemistry. / Volume 4**

**Example text**

Theorem 5 . The operators B and C are bounded, their bounds do not exceed one. B is a positive self-adjoint operator, C = AB, and (Z+ A'A)B = Z. (IV. 14) Bela A. 13) can be solved for any h ensures that the operator I + AtA has an inverse. ijwe have f = Bh and ( I + A t A ) f = (I+AtA)Bh = h. 14). 10) splits the element {h, 0}in two orthogonal parts, therefore llh1I2 = I Ih 0}1l2= I l f 112 + llAf 112 + IIAfgll2 + 11g1I2. 15) Hence llBh1I2 + IICh1I2 = I l f 112 + 1191125 llh1I2, (IV. 16) therefore the bounds of B and C cannot exceed one.

42) also follows by induction with the aid of the equation Yn+l- Y,,= +[ Y,,’ - Y,Z-,] = $( Y,, - Y,,-I)( Y,, + Y,,-i). 47) Beta A. Lengyel 40 Here we make use of the fact that the product of positive operators is positive. By virtue of their construction the Y’s are permutable with every bounded operator permutable with H. The existence of a limiting operator Y = lim Y, is now assured. Moreover 0 < Y < I. 45) in the following form: H 2 = I - 2Y,+, + Y,”. 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.

3. Prove that the operator D on Q2[0,a33 is a closed symmetric operator. 4. Show that the operator d2/dx2with a proper definition of its domain is self-adjoint in all three types of 2, spaces discussed. E. An Auxiliary Theorem about Bounded Self-Adjoint Operators In the proof of the spectral theorem for general self-adjoint operators we shall need a theorem concerning bounded self-adjoint operators. This theorem asserts that if such an operator is bounded away from zero then it Bela A. Lengyel 58 can be inverted and the inverse is again a bounded self-adjoint operator.

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