Consider the differential operator $D:$
$$
Du:=\frac{-d^2}{dx^2}u
$$
on the function space $$C=\{ u\in C^2([0,1]):u(0)=u(1)=0\}.$$


It's not hard to find the eigenvalues and eigenvectors(eigenfunctions) for $D$ by soloving the eigenvalue problem:
$$
-u''=\lambda u\qquad u(0)=u(1)=0.
$$


Here are my **questions**:



> 
> * For each of the differential operators $D^m(m=2,3,4,\dots)$, what boundary conditions should one choose to ensure that $D^m$ and $D$ share the *same* eigenfunctions? I have no idea for even $m=2$.
> * What if 
>  $$
> Du:=\frac{-d^2}{dx^2}u+u?
> $$
> 
> 
> 



Choosing boundary conditions for $(\frac{-d^2}{dx^2})^m$ on $H^m((0,1))$?
This is easy. If the boundary condition for D is, say, $Lu=0$, then for $D^m$ you should pose the boundary conditions
$$Lu=LDu=LD^2u=...=LD^{m-1}u=0.$$


