Editing Self-adjoint operator (section)

=== Pure point spectrum ===
{{distinguish|Discrete spectrum (mathematics)}}
A self-adjoint operator ''A'' on ''H'' has pure [[point spectrum]] if and only if ''H'' has an orthonormal basis {''e<sub>i</sub>''}<sub>''i'' ∈ I</sub> consisting of eigenvectors for ''A''.

'''Example'''. The Hamiltonian for the harmonic oscillator has a quadratic potential ''V'', that is
: <math>-\Delta  + |x|^2.</math>

This Hamiltonian has pure point spectrum; this is typical for bound state [[Hamiltonian (quantum mechanics)|Hamiltonians]] in quantum mechanics.{{clarify|reason=The above example is from classical mechanics|date=November 2023}}{{sfn | Ruelle | 1969|ps=}} As was pointed out in a previous example, a sufficient condition that an unbounded symmetric operator has eigenvectors which form a Hilbert space basis is that it has a compact inverse.