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

The theorem is named after Ernst David Hellinger and Otto Toeplitz.

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.

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) 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).