Question

Find the eigenfunctions and their associated eigenvalues of the operator [-d/dx^2]

Answer 1

\[-\frac{d}{dx^2}\]

Answer 2

Eigenfunctions are \[e^{\sigma x}, Cos{(\nu x + \theta)}\]

Answer 3

Here the three parameters are CONSTANTS
\[\sigma, \nu, \theta\]
The first function's eigenvalue will be -sigma^2, the second case eigenvalue will be
\[\nu^2\]
B.t.w. the sin is of course a special case of the \[Cos(\nu x + \theta)\]

Answer 4

@henpen

Answer 5

Ahh, and one minor correction to your question: one should write of course\[\frac{ d^2 }{ dx^2 }\]
and not the way u wrote.

Answer 6

Thank you! You are a beautiful openstudier!

Answer 7

Is there any proof that there can be no other functions that 'fit'?

Answer 8

Please write it as "testimonial" I will be very much obliged to you.

Answer 9

Yes - there is a proof - it involves the fact that this is a Shturm-Liouville operator and these function s form a COMPLETE BASE for ALL possible solutions.

For Details consult - Shturm-Liouville theory, Second Order Equations, Hilbert Sapce BAses

Answer 10

'Hilbert Space Bases' are a single thing, or is it Hilbert space and bases?

Answer 11

Well try both. But As a single thing - if works, is the thing

Answer 12

Which of your list holds the simplest proof?

Answer 13

Shturm-Liouville theory