Infunctional analysis, a branch ofmathematics, theHellinger–Toeplitz theorem states that an everywhere-definedsymmetric operator on aHilbert space withinner product${\displaystyle ⟨\cdot |\cdot ⟩}$ isbounded. By definition, an operatorA issymmetric if

${\displaystyle ⟨Ax|y⟩=⟨x|Ay⟩}$

for allx,y in the domain ofA. Note that symmetriceverywhere-defined operators are necessarilyself-adjoint, so this theorem can also be stated as follows: an everywhere-defined self-adjoint operator is bounded. The theorem is named afterErnst David Hellinger andOtto Toeplitz.

This theorem can be viewed as an immediate corollary of theclosed graph theorem, as self-adjoint operators areclosed. Alternatively, it can be argued using theuniform 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 operatorA is defined everywhere (and, in turn, the completeness of Hilbert spaces).

The Hellinger–Toeplitz theorem reveals certain technical difficulties in themathematical 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 adense subset). Take for instance thequantum harmonic oscillator. Here the Hilbert space isL²(R), the space of square integrable functions onR, and the energy operatorH is defined by (assuming the units are chosen such that ℏ = m = ω = 1)

${\displaystyle [Hf](x)=-\frac{1}{2}\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}f(x)+\frac{1}{2}{x}^{2}f(x).}$

This operator is self-adjoint and unbounded (itseigenvalues are 1/2, 3/2, 5/2, ...), so it cannot be defined on the whole of L²(R). In other words, it will map some functions in L²(R) to functions that are no longer square integrable. One such function could be

${\displaystyle \psi (x)=\frac{1}{\pi \sqrt{1+x^2}}.}$

• Reed, Michael andSimon, 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.

• Theorems in functional analysis
• Hilbert spaces

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