Consider a second-order differential operator
(1)
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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)
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(3)
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In order for the operator to be self-adjoint, i.e.,
(4)
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the second terms in (◇) and (◇) must be equal, so
(5)
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This also guarantees that the third terms are equal, since
(6)
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so (◇) becomes
(7)
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(8)
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(9)
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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)
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(11)
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(12)
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(13)
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where is a constant of integration.
A self-adjoint operator which satisfies the boundary conditions
(14)
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is automatically a Hermitian operator.