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:
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—Henry David Thoreau (1817–1862)
“The point is to show who is the cross and who the crucified.”
—Max Frisch (1911–1991)