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

sameeigenfunctions? I have no idea for even $m=2$.What if $$ Du:=\frac{-d^2}{dx^2}u+u? $$

`math.SE`

for several days without an answer. – Jack May 3 '12 at 16:08