Lysario – by Panagiotis Chatzichrisafis

"ούτω γάρ ειδέναι το σύνθετον υπολαμβάνομεν, όταν ειδώμεν εκ τίνων και πόσων εστίν …"

Archive for Januar, 2009

The exercise [1, p. 86 ex. 3.16] asks to prove that if the eigenfunctions and eigenvalues of an operator  \widehat{A} are \{\phi_{n}\} and \{a_{n}\}, respectively (\ensuremath{\widehat{A}}\phi_{n}=a_{n}\phi_{n}) then the eigenfunctions of a function f(x) having an expansion of the form:
f(x)=\sum\limits^{\infty}_{l=0}b_{l}x^{l} (1)

will also be \phi_n  with corresponding eigenvalues f(a_n) , n=0,1... . That is f(\widehat{A})\phi_n=f(a_n)\phi_n. read the conclusion >
Exercise [1, p. 14 exercise 1.1] asks to prove that
g(x)=\frac{1}{1-x} (1)

is a monotonically increasing function over the interval 0<x<1. Furthermore it asks to show that due to the monotonicity of the peaks of the spectral estimate shown in  [1, Figure 1.2b] that they must also be present in the spectral estimate of [1, Figure 1.2a]. read the conclusion >

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