Deficiency indices - Knowlpedia

Let $A$ be a densely defined closed symmetric operator on a complex Hilbert space. Its deficiency subspaces are

\mathcal N_+=\ker(A^*-iI),\qquad \mathcal N_-=\ker(A^*+iI),

where $A^*$ is the adjoint of $A$ . The deficiency indices are the cardinal dimensions

n_+(A)=\dim\mathcal N_+,\qquad n_-(A)=\dim\mathcal N_-.

They may be finite or infinite. More generally, $\dim\ker(A^*-zI)$ is constant as $z$ ranges over either open half-plane; $n_+$ uses the upper half-plane and $n_-$ the lower half-plane under the convention above.

Self-adjointness and extensions

The operator $A$ is self-adjoint exactly when $n_+(A)=n_-(A)=0$ . It has a self-adjoint extension on the same Hilbert space exactly when $n_+(A)=n_-(A)$ ; when these common dimensions are nonzero, unitary maps from $\mathcal N_+$ to $\mathcal N_-$ parametrize the extensions. This is von Neumann’s extension theorem.

Closure and geometric meaning

If a symmetric operator is not closed, its deficiency indices are defined to be those of its closure. For a closed symmetric $A$ , the domain of $A^*$ decomposes into the domain of $A$ together with the two deficiency subspaces. The latter measure the independent boundary data missing from $A$ ; equality of their dimensions is precisely what permits those data to be paired to form a self-adjoint domain.

Examples and sign convention

The minimal momentum operator $-i\,d/dx$ on a bounded interval, obtained by closing the operator on compactly supported smooth functions, has deficiency indices $(1,1)$ and admits a circle of self-adjoint boundary conditions. On the whole real line the corresponding minimal operator has indices $(0,0)$ and is essentially self-adjoint. Interchanging $i$ and $-i$ , or changing the sign of the operator, swaps the ordered pair; a source’s convention should therefore be checked before comparing formulas.

References

Konrad Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Graduate Texts in Mathematics 265, Springer, 2012. DOI record. Relevant: Chapter 13 on deficiency indices and self-adjoint extensions.
Michael Reed and Barry Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, 1975. Bibliographic record. Relevant: Chapter X on self-adjoint extensions.