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GeezerVison is showing mathworld.wolfram.com — Self-Adjoint -- from Wolfram MathWorld

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Self-Adjoint

A densely defined operator on a Hilbert space is self-adjoint if it equals its adjoint operator, including equality of their domains. For a differential operator, this condition must be distinguished from formal self-adjointness of the differential expression. Consider the second-order expression

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 formal adjoint L^~^| is defined by

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

In order for the differential expression to be formally self-adjoint, i.e.,

L^~=L^~^|,
(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)=>p_0^('')(x)=p_1^'(x),
(5)

so (2) becomes

L^~u=L^~^|u
=p_0(d^2u)/(dx^2)+p_0^'(du)/(dx)+p_2u
=d/(dx)(p_0(du)/(dx))+p_2u.
(6)

The differential operators corresponding to the Legendre differential equation and the equation of simple harmonic motion are formally self-adjoint, while those corresponding to the Laguerre differential equation and Hermite differential equation are not formally self-adjoint in their displayed forms.

A second-order ordinary differential equation of the form above can be put in formally self-adjoint form using an integrating factor, yielding a Sturm-Liouville equation. For the homogeneous equation L^~u=0 in the special case p_2(x)=0, (6) gives

d/(dx)[p_0(x)(du)/(dx)]=0
(7)
p_0(x)(du)/(dx)=C
(8)
du=C(dx)/(p_0(x))
(9)
u=Cint(dx)/(p_0(x)),
(10)

where C is a constant of integration.

A choice of domain and boundary conditions is additionally required to obtain a self-adjoint operator. For a Sturm-Liouville expression, suitable boundary conditions make the boundary form

[v^_p_0u^'-p_0v^_^'u]_a^b=0.
(11)

Formal self-adjointness alone does not imply self-adjointness of the associated operator.

See also

Adjoint Operator, Formal Adjoint, Hermitian Operator, Sturm-Liouville Equation

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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.Conway, J. B. A Course in Functional Analysis. New York: Springer-Verlag, 1990.

Referenced on Wolfram|Alpha

Self-Adjoint

Cite this as:

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

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