Let $\displaystyle m\in\mathbb{N}$.

Consider the following boundary value problem

$\displaystyle \begin{cases}L\psi(n)+\lambda\psi(n)=0,&n\in\{1,2, \ldots,m\}\\ \psi(n)=0,&n\in\{0,m+1\},\end{cases}\rule{4cm}{0cm }(\star)$

where the operator $\displaystyle L$ is defined to be

$\displaystyle L\psi(n):=\Delta^{2}\psi(n-1)$ and $\displaystyle \Delta$ is the usual forward difference operator.

When I try to find a nontirivial real sequence satisfying $\displaystyle (\star)$ with $\displaystyle \lambda>0$ and $\displaystyle m=2$ or $\displaystyle m=3$, I always fail.

I need help to find a suitable $\displaystyle \lambda>0$ and $\displaystyle m\in\mathbb{N}$ for which $\displaystyle (\star)$ admits a nontrivial solution $\displaystyle \psi$.

Thanks.