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for all x, y in the domain of A. Note that symmetric everywhere defined operators are necessarily self-adjoint, that is A = A*. Unbounded symmetric operators (particularly self-adjoint ones) are plentiful and in fact are extremely important in applications to physics.
The Hellinger-Toeplitz theorem leads to some 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. 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)
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).
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