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Self-Adjoint


Consider a second-order differential operator

 L^~u(x)=p_0(d^2u)/(dx^2)+p_1(du)/(dx)+p_2u,
(1)

where u=u(x) and p_i=p_i(x) are real functions of x on the region of interest [a,b] with 2-i continuous derivatives and with p_0(x)!=0 on [a,b]. This means that there are no singular points in [a,b]. Then the adjoint operator L^~^| is defined by

L^~^|u=(d^2)/(dx^2)(p_0u)-d/(dx)(p_1u)+p_2u
(2)
=p_0(d^2u)/(dx^2)+(2p_0^'-p_1)(du)/(dx)+(p_0^('')-p_1^'+p_2)u.
(3)

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

 L^~=L^~^|,
(4)

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

 p_0^'(x)=p_1(x).
(5)

This also guarantees that the third terms are equal, since

 p_0^'(x)=p_1(x)=>p_0^('')(x)=p_1^'(x),
(6)

so (◇) becomes

L^~u=L^~^|u
(7)
=p_0(d^2u)/(dx^2)+p_0^'(du)/(dx)+p_2u
(8)
=d/(dx)(p_0(du)/(dx))+p_2u=0.
(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 p_2(x)=0, (9) gives

 d/(dx)[p_0(x)(du)/(dx)]=0
(10)
 p_0(x)(du)/(dx)=C
(11)
 du=C(dx)/(p_0(x))
(12)
 u=Cint(dx)/(p_0(x)),
(13)

where C is a constant of integration.

A self-adjoint operator which satisfies the boundary conditions

 v^_pU^'|_(x=a)=v^_pU^'|_(x=b)
(14)

is automatically a Hermitian operator.


See also

Adjoint, Hermitian Operator, Sturm-Liouville Theory

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References

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

Referenced on Wolfram|Alpha

Self-Adjoint

Cite this as:

Weisstein, Eric W. "Self-Adjoint." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Self-Adjoint.html

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