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 not naïve enough to ask for pure men; we ask merely for men whose impurity does not conflict with the obligations of their job.”
—Jean Rostand (1894–1977)
“What I am anxious to do is to secure my legislation.... What I want to do is to get through that, and if I can point to a record of usefulness of that kind, I am entirely willing to quit office.”
—William Howard Taft (1857–1930)