Hellinger–Toeplitz theorem

In functional analysis, a branch of mathematics, the Hellinger–Toeplitz theorem states that an everywhere-defined symmetric operator on a Hilbert space with inner product $\langle \cdot | \cdot \rangle$ is bounded. By definition, an operator A is symmetric if

\langle A x | y \rangle = \langle x | A y\rangle

for all x, y in the domain of A. Note that symmetric everywhere-defined operators are necessarily self-adjoint, so this theorem can also be stated as follows: an everywhere-defined self-adjoint operator is bounded. The theorem is named after Ernst David Hellinger and Otto Toeplitz.

This theorem can be viewed as an immediate corollary of the closed graph theorem, as self-adjoint operators are closed. Alternatively, it can be argued using the uniform boundedness principle. One relies on the symmetric assumption, therefore the inner product structure, in proving the theorem. Also crucial is the fact that the given operator A is defined everywhere (and, in turn, the completeness of Hilbert spaces).

The Hellinger–Toeplitz theorem reveals certain technical difficulties in the mathematical formulation of quantum mechanics. Observables in quantum mechanics correspond to self-adjoint operators on some Hilbert space, but some observables (like energy) are unbounded. By Hellinger–Toeplitz, such operators cannot be everywhere defined (but they may be defined on a dense subset). Take for instance the quantum harmonic oscillator. Here the Hilbert space is L²(R), the space of square integrable functions on R, and the energy operator H is defined by (assuming the units are chosen such that ℏ = m = ω = 1)

[Hf](x) = - \frac12 \frac{\mathrm{d}^2}{\mathrm{d}x^2} f(x) + \frac12 x^2 f(x).

This operator is self-adjoint and unbounded (its eigenvalues are 1/2, 3/2, 5/2, ...), so it cannot be defined on the whole of L²(R).

References

Reed, Michael and Simon, Barry: Methods of Mathematical Physics, Volume 1: Functional Analysis. Academic Press, 1980. See Section III.5.
Teschl, Gerald (2009). Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators. Providence: American Mathematical Society. ISBN 978-0-8218-4660-5.

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