Advances in Quantum Chemistry Volume 4-1968 by Per-Olov Lowdin

By Per-Olov Lowdin

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Whether the graph of A is closed, or not, the linear manifold orthogonal to {f,Af} is closed, therefore the set {Atg, - g } is a closed set. Consequently the set { g , Atg} obtained by the application of the norm-preserving operators S and U is also closed. Whenever the adjoint operator At exists, it is a closed operator. When the operator A has closed extensions, the least of these is determined by adding the limit points to the linear manifold {f, Af}. This process is equivalent to determining the orthogonal complement of the closed linear manifold formed by the pairs { A t g , - g } .

Therefore ll~f1I2 = ( H Z f ,f ) = j;, 12 4E,f,f). 30) We have now obtained a new proof of the fact that llHf1I2 S K2Jlf1I2,or IlHfll 5 Kllfll. 31) 37 Functional Analysis for Quantum Theorists valid for any positive integer n. It may be proven by mathematical induction. 33) + + + valid for every$ When the polynomial Pn(A)= a, a l l ... a&" has real coefficients, the operator P(H) is self-adjoint. Its bound does not exceed the maximum of IP,(A)l in the interval - K to K. + C. Auxiliary Theorems of F.

The solutions of the Eq. 3) contain a complete orthonormal system of vectors. When these vectors are introduced to define the axes of a coordinate system in the n-dimensioned space, the Hermitean matrix assumes the following diagonal form : 1 ; j. Anumber occurring more than once is said to be a multiple, or degenerate eigenvalue. At first one might think that it ought to be possible to generalize the spectral theorem to the infinite-dimensional case by simply permitting an infinite number of eigenvalues and eigenfunctions, and that one ought to obtain a diagonal matrix with infinitely many rows and columns.

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