Abstract
We remind the reader of the basics of the general theory of self-adjoint extensions of unbounded symmetric operators. The exposition is based on a notion of asymmetry forms of the adjoint operator. The principal statements concerning the possibility and specification of self-adjoint extensions both in terms of isometries between the deficient subspaces and in terms of the asymmetry forms are collected in the main theorem, followed by a comment on a direct application of the main theorem to physical problems of quantization.
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- 2.
We point out that there exists an anticorrespondence \(z \rightleftarrows \bar{ z}\) between the subscript z of ℵ z and the respective eigenvalue \(\bar{z}\) and the subscript of the eigenvector \({\xi }_{\bar{z}}\) of \(\hat{{f}}^{+}\). Perhaps it would be more convenient to change the notation \({\aleph }_{z} \rightleftarrows {\aleph }_{\bar{z}}\); the conventional notation is due to tradition. The same concerns the subscripts of m ± and \({\mathbb{C}}_{\mp }\).
- 3.
Although \({\aleph }_{\overline{z}}\) and ℵ z are closed subspaces in \(\mathfrak{H}\), we cannot generally assert that their direct sum \({\aleph }_{\overline{z}} + {\aleph }_{z}\) is also a closed subspace. The latter is always true if one of the subspaces is finite-dimensional.
- 4.
This is well known to physicists as applied to s.a. operators.
- 5.
We can omit this requirement because an isometric operator is linear [9].
- 6.
Under our agreement that \(\dim {\aleph }_{\overline{z}} \leq \dim {\aleph }_{z}\).
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Provided, for example, by canonical quantization rules for classical observables f(q, p).
- 8.
Self-adjoint by Lagrange.
- 9.
We would like to emphasize that at this point, the general theory requires evaluating the closure \(\overline{\hat{f}}\). It is precisely \(\overline{\hat{f}}\) and \({D}_{\overline{f}}\) that enter (3.31), (3.32), (3.34), and (3.35), while in the physics literature we can sometimes see that in citing and using these formulas, \(\hat{f}\) and D f stand for \(\overline{\hat{f}}\) and \({D}_{\overline{f}}\) even for a nonclosed symmetric operator \(\hat{f}\), which is incorrect.
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Gitman, D.M., Tyutin, I.V., Voronov, B.L. (2012). Basics of the Theory of Self-adjoint Extensions of Symmetric Operators. In: Self-adjoint Extensions in Quantum Mechanics. Progress in Mathematical Physics, vol 62. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4662-2_3
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DOI: https://doi.org/10.1007/978-0-8176-4662-2_3
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