Pure Point Spectrum
A self-adjoint operator A on H has pure point spectrum if and only if H has an orthonormal basis {ei}i ∈ I consisting of eigenvectors for A.
Example. The Hamiltonian for the harmonic oscillator has a quadratic potential V, that is
This Hamiltonian has pure point spectrum; this is typical for bound state Hamiltonians in quantum mechanics. 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.
Read more about this topic: Self-adjoint Operator
Famous quotes containing the words pure and/or point:
“The Family is the Country of the heart. There is an angel in the Family who, by the mysterious influence of grace, of sweetness, and of love, renders the fulfilment of duties less wearisome, sorrows less bitter. The only pure joys unmixed with sadness which it is given to man to taste upon earth are, thanks to this angel, the joys of the Family.”
—Giuseppe Mazzini (1805–1872)
“The striking point about our model family is not simply the compete-compete, consume-consume style of life it urges us to follow.... The striking point, in the face of all the propaganda, is how few Americans actually live this way.”
—Louise Kapp Howe (b. 1934)