info prev up next book cdrom email home

Self-Adjoint Operator

Given a differential equation

{\tilde {\mathcal L}} u(x) \equiv p_0 {du^2 \over dx^2} +p_1 {du \over dx} +p_2 u,
\end{displaymath} (1)

where $p_i \equiv p_i(x)$ and $u\equiv u(x)$, the Adjoint Operator ${\tilde {\mathcal L}}^\dagger$ is defined by

{\tilde {\mathcal L}}^\dagger u \equiv {d \over dx^2} (p_0 u... dx^2} +(2{p_0}'-p_1) {du \over dx} + ({p_0}''-{p_1}'+p_2)u.
\end{displaymath} (2)

In order for the operator to be self-adjoint, i.e.,
{\tilde {\mathcal L}} = {\tilde {\mathcal L}}^\dagger,
\end{displaymath} (3)

the second terms in (1) and (2) must be equal, so
\end{displaymath} (4)

This also guarantees that the third terms are equal, since
{p_0}'(x)=p_1(x) \Rightarrow {p_0}''(x)={p_1}'(x),
\end{displaymath} (5)

so (2) becomes
$\displaystyle {\tilde {\mathcal L}}u$ $\textstyle =$ $\displaystyle {\tilde {\mathcal L}}^\dagger u= p_0 {d^2 \over dx^2} + {p_0}' {du \over dx}+p_2 u$ (6)
  $\textstyle =$ $\displaystyle {d \over dx}\left({p_0 {du \over dx}}\right)+p_2 u=0.$ (7)

The Legendre Differential Equation and the equation of Simple Harmonic Motion are self-adjoint, but the Laguerre Differential Equation and Hermite Differential Equation are not.

A nonself-adjoint second-order linear differential operator can always be transformed into a self-adjoint one using Sturm-Liouville Theory. In the special case $p_2(x)=0$, (7) gives

{d\over dx}\left[{p_0(x) {du\over dx}}\right]= 0
\end{displaymath} (8)

p_0(x){du\over dx} = C
\end{displaymath} (9)

du = C {dx\over p_0(x)}
\end{displaymath} (10)

u = C \int{dx\over p_0(x)},
\end{displaymath} (11)

where $C$ is a constant of integration.

A self-adjoint operator which satisfies the Boundary Conditions

v^*pU'\vert _{x=a} = v^*pU'\vert _{x=b}
\end{displaymath} (12)

is automatically a Hermitian Operator.

See also Adjoint Operator, Hermitian Operator, Sturm-Liouville Theory


Arfken, G. ``Self-Adjoint Differential Equations.'' §9.1 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 497-509, 1985.

info prev up next book cdrom email home

© 1996-9 Eric W. Weisstein