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:
“We are born with luck
which is to say with gold in our mouth.
As new and smooth as a grape,
as pure as a pond in Alaska,
as good as the stem of a green bean
we are born and that ought to be enough....”
—Anne Sexton (1928–1974)
“Bias, point of view, fury—are they ... so dangerous and must they be ironed out of history, the hills flattened and the contours leveled? The professors talk ... about passion and point of view in history as a Calvinist talks about sin in the bedroom.”
—Catherine Drinker Bowen (1897–1973)