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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 is my question:

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 [EDIT: set of] eigenfunctions?

Added(Thanks to Michael Renardy): Denote the boundary conditions for $D$ as $Lu=0$ and suppose that $Du=\lambda u$. Then we have $$ D^{m}u=\lambda^mu $$ and $$ L(D^ku)=L(\lambda^k u)=\lambda^k Lu=0\quad k =1,2,\cdots,m-1 $$ Now the problem reduce to show that if $D^mu=\lambda^m u$ and $$ L(D^ku)=0\quad k =1,2,\cdots,m-1 $$ then $Du=\lambda u$.

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