Self-adjoint Operator - Examples

Examples

We first consider the differential operator

defined on the space of complex-valued C∞ functions on vanishing near 0 and 1. D is a symmetric operator as can be shown by integration by parts. The spaces N+, N− are given respectively by the distributional solutions to the equation

which are in L2 . One can show that each one of these solution spaces is 1-dimensional, generated by the functions x → eix and x → e−ix respectively. This shows that D is not essentially self-adjoint, but does have self-adjoint extensions. These self-adjoint extensions are parametrized by the space of unitary mappings

which in this case happens to be the unit circle T.

This simple example illustrates a general fact about self-adjoint extensions of symmetric differential operators P on an open set M. They are determined by the unitary maps between the eigenvalue spaces

where Pdist is the distributional extension of P.

We next give the example of differential operators with constant coefficients. Let

be a polynomial on Rn with real coefficients, where α ranges over a (finite) set of multi-indices. Thus

and

We also use the notation

Then the operator P(D) defined on the space of infinitely differentiable functions of compact support on Rn by

is essentially self-adjoint on L2(Rn).

Theorem. Let P a polynomial function on Rn with real coefficients, F the Fourier transform considered as a unitary map L2(Rn) → L2(Rn). Then F* P(D) F is essentially self-adjoint and its unique self-adjoint extension is the operator of multiplication by the function P.

More generally, consider linear differential operators acting on infinitely differentiable complex-valued functions of compact support. If M is an open subset of Rn

where aα are (not necessarily constant) infinitely differentiable functions. P is a linear operator

Corresponding to P there is another differential operator, the formal adjoint of P

Theorem. The operator theoretic adjoint P* of P is a restriction of the distributional extension of the formal adjoint. Specifically:

\operatorname{dom} P^* = \{u \in L^2(M): P^{\mathrm{*form}}u \in L^2(M)\}.

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