Spectral decomposition, Sturm-Liouville

Question

Consider $A:L^2([0,1]) \to L^2([0,1])$ defined as $$ (Af)(s) = \int_0^1 \max\{s,t\}\cdot f(t)\mathrm dt$$ I've shown that the operator is compact and self-adjoint.

Looking for the eigenvalues.

$$(Af)(s) = cf(s) = \int_0^1 \max\{s,t\}\cdot f(t)\mathrm dt$$ then $$c\cdot \frac{\mathrm df}{\mathrm ds} = \int_0^s f(t)\mathrm dt$$ and $$c\cdot \frac{\mathrm d^2f}{\mathrm ds^2} = f(s).$$

Obviously $$\frac{\mathrm df(0)}{\mathrm ds} = 0.$$

How can I get another initial condition?

Please rember that Latex cannot be considered an optional on this website. — uforoboa, Jan 23 '13 at 12:38

score 2 · Accepted Answer · 2013-01-23T15:01:34.037

2

The other condition occurs at the other endpoint: $$ c f(1) = \int_0^1 f(t)\,dt = cf'(1) $$ hence $f(1)=f'(1)$.

The general method for finding boundary conditions in such problems is to write down $f(0),f'(0),f(1),f'(1)$ and to stare at them.

edited Jan 23 '13 at 15:01

answered Jan 23 '13 at 14:52

Yes, thank you. – superAnnoyingUser Jan 23 '13 at 20:16
@George On the technical side, you may be interested in this: How do I accept an answer? – Jan 23 '13 at 20:25

Spectral decomposition, Sturm-Liouville

1 Answers1