 Sturm-Liouville problem

A regular Sturm-Liouville problem specifies a differential equation, a positive weight, and boundary conditions together. In one standard sign convention,For example, take real continuous $q,w$, continuously differentiable $p$, and $p,w>0$ on the closed bounded interval. Separated homogeneous self-adjoint endpoint conditions can be written $A_{a}y(a)+B_a(p{y}^{\prime })(a)=0$ and $A_{b}y(b)+B_b(p{y}^{\prime })(b)=0$, with each real coefficient pair nonzero. The parameter values admitting nonzero solutions are its eigenvalues; the solutions are its Sturm-Liouville eigenfunctions. The boundary conditions are essential: the differential equation alone is not an eigenvalue problem.^[ref]

For $p=w=1$, $q=0$ and zero endpoint values on $(0,\pi )$, the eigenvalues are $n^2$ and eigenfunctions are $\sin (nx)$, $n\ge 1$. Replacing the endpoints by derivative-zero conditions admits the constant eigenfunction at eigenvalue zero. Thus changing the boundary conditions changes the spectrum without changing the interior equation.