Self-Adjoint

Consider a second-order differential operator

(1)

where and are real functions of on the region of interest with continuous derivatives and with on . This means that there are no singular points in . Then the adjoint operator is defined by

			(2)
			(3)

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

(4)

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

(5)

This also guarantees that the third terms are equal, since

(6)

so (◇) becomes

			(7)
			(8)
			(9)

The differential operators corresponding to the Legendre differential equation and the equation of simple harmonic motion are self-adjoint, while those corresponding to 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 , (9) gives

(10)

(11)

(12)

(13)

where is a constant of integration.

A self-adjoint operator which satisfies the boundary conditions

(14)

is automatically a Hermitian operator.