Self-adjoint fourth-order scalar differential operator
= Self-adjoint fourth-order scalar differential operator
{title2=$D^4+pD^2+p'D+r$}
For real coefficient functions, the fourth-order scalar operator
$$
L=D^4+pD^2+qD+r
$$
is formally self-adjoint exactly when $q=p'$. Indeed, $(pD^2)^*=pD^2+2p'D+p''$ and $(qD)^*=-qD-q'$. Under <clamped boundary condition>[clamped boundary conditions], its quadratic form is
$$
\langle y,Ly\rangle=\int\left[(y'')^2-p(y')^2+ry^2\right]dx.
$$