Self-Adjoint Operator

Given a differential equation

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

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

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

In order for the operator to be self-adjoint, i.e.,

$${\tilde {\mathcal L}} = {\tilde {\mathcal L}}^\dagger,$$ (3)

the second terms in (1) and (2) must be equal, so

$${p_0}'(x)=p_1(x).$$ (4)

This also guarantees that the third terms are equal, since

$${p_0}'(x)=p_1(x) \Rightarrow {p_0}''(x)={p_1}'(x),$$ (5)

so (2) becomes

$${\tilde {\mathcal L}}u = {\tilde {\mathcal L}}^\dagger u= p_0 {d^2 \over dx^2} + {p_0}' {du \over dx}+p_2 u$$ (6)

$$= {d \over dx}\left({p_0 {du \over dx}}\right)+p_2 u=0.$$ (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

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

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

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

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

where $C$ is a constant of integration.

A self-adjoint operator which satisfies the Boundary Conditions

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

is automatically a Hermitian Operator.

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

References

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