# 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. 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 >