Existence of spectral representation for unbounded symmetric operators

Establish the existence of a spectral representation (a partition of unity E(λ) such that T = ∫_{−∞}^{+∞} λ dE(λ)) for general unbounded symmetric (self-adjoint) operators on a Hilbert space, beyond the known case of bounded operators, to cover the unbounded operators predominant in quantum mechanics such as position, momentum, and Hamiltonian operators.

Background

The paper reviews the spectral theory available in 1927: Hilbert–Hellinger theory guarantees a spectral representation for bounded symmetric operators. However, many quantum-mechanical observables (e.g., position, momentum, energy) are represented by unbounded operators, and at the time it was unsettled whether a comparable spectral resolution existed for them.

Von Neumann later provided a general solution, but within the 1927 context this was explicitly identified as an unresolved issue central to establishing a rigorous mathematical foundation for quantum mechanics.