1. ## [SOLVED] eigenvalue problem

I want to find the eigenvalues and the eigenfunctions of the operator

$\displaystyle ({d^{2}}/{dx^{2}})+k$

so we have the problem

$\displaystyle ({d^{2}y_{n}}/{dx^{2}})+ky_{n}=L_{n} y_{n}$

Now I put in the functie y=A*exp(n*x), is it correct to do this?, and get the equation

$\displaystyle n^{2}+k-L_{n} =0$

so Ln=n^2+k

but acording to the solutions it has to be Ln=n^2-k. Where do I make the mistake?
and how do I exactly determine the eigenfunctions

thanks

2. Originally Posted by charlie123
I want to find the eigenvalues and the eigenfunctions of the operator

$\displaystyle ({d^{2}}/{dx^{2}})+k$

so we have the problem

$\displaystyle ({d^{2}y_{n}}/{dx^{2}})+ky_{n}=L_{n} y_{n}$

Now I put in the functie y=A*exp(n*x), is it correct to do this?, and get the equation

$\displaystyle n^{2}+k-L_{n} =0$

so Ln=n^2+k

but acording to the solutions it has to be Ln=n^2-k. Where do I make the mistake?
and how do I exactly determine the eigenfunctions
This problem as stated is incomplete. The differential operator is not completely defined unless some boundary conditions are specified. If the boundary condition says that y→∞ as x→∞ then you are correct to take functions of the form A*exp(n*x) as eigenfunctions. More often, the boundary conditions will say for example y=0 when x = 0 and when x = π. In that case, the eigenfunctions will be of the form A*sin(nx) and the eigenvalues will be k–n^2 (still not equal to the given solution n^2–k, but perhaps you are using a different convention about signs).

3. the boundary conditions are y(0)=0 and y(pi)=0

but how do I know what the the eigenfunction is? Here I can see it of course, but if I didn't know it for instance how do I start solving this problem?
The determining of the eigenvalue is not difficult after that.

thanks for the help

4. Originally Posted by charlie123
the boundary conditions are y(0)=0 and y(pi)=0

but how do I know what the the eigenfunction is? Here I can see it of course, but if I didn't know it for instance how do I start solving this problem?
The determining of the eigenvalue is not difficult after that.
The differential equation is $\displaystyle d^2y/dx^2 + (k-\lambda)y = 0$, where $\displaystyle \lambda$ is an eigenvalue. This is a standard SHM equation, and I guess you're supposed to know that the general solution is $\displaystyle y = A\sin\omega x + B\cos\omega x$, where $\displaystyle \omega^2 = k-\lambda$. The boundary condition y(0)=0 tells you that B=0, and the boundary condition y(π)=0 tells you that $\displaystyle \omega$ must be an integer, say n. Therefore $\displaystyle k-\lambda = n^2$.

So you see it's not a question of finding the eigenfunction first and then the eigenvalue. They both emerge together as a result of solving the differential equation together with its boundary conditions.