distributional derivative
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21: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►of the Dirac delta distribution.
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►For to be actually self adjoint it is necessary to also show that , as it is often the case that and have different domains, see Friedman (1990, p 148) for a simple example of such differences involving the differential operator .
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►The special form of (1.18.28) is especially useful for applications in physics, as the connection to non-relativistic quantum mechanics is immediate: being proportional to the kinetic energy operator for a single particle in one dimension, being proportional to the potential energy, often written as , of that same particle, and which is simply a multiplicative operator.
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►Possible eigenfunctions of being , , , consider three cases, which illustrate the importance of boundary conditions.
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►For a formally self-adjoint second order differential operator , such as that of (1.18.28), the space can be seen to consist of all such that the distribution
can be identified with a function in , which is the function .
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22: Errata
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►The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions.
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Equations (10.15.1), (10.38.1)
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Subsection 1.16(vii)
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Subsection 1.16(viii)
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Equation (31.12.3)
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These equations have been generalized to include the additional cases of , , respectively.
Several changes have been made to
An entire new Subsection 1.16(viii) Fourier Transforms of Special Distributions, was contributed by Roderick Wong.
31.12.3
Originally the sign in front of the second term in this equation was . The correct sign is .
Reported 2013-10-31 by Henryk Witek.