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:
“Whenever a person strives, by the help of dialectic, to start in pursuit of every reality by a simple process of reason, independent of all sensuous information—never flinching, until by an act of the pure intelligence he has grasped the real nature of good—he arrives at the very end of the intellectual world.”
—Plato (c. 427–347 B.C.)
“People talk about the conscience, but it seems to me one must just bring it up to a certain point and leave it there. You can let your conscience alone if you’re nice to the second housemaid.”
—Henry James (1843–1916)