In several cases, analytic results can be obtained more easily in the study of one-dimensional systems. For a quantum particle with a wave function | ψ ψ ⟩ ⟩ moving in a one-dimensional potential V ( x ) , the time-independent Schrödinger equation can be written as − − ℏ ℏ 2 2 m d 2 ψ ψ d x 2 + V ψ ψ = E ψ ψ Since this is an ordinary differential equation, there are two independent eigenfunctions for a given energy E at most, so that the degree of degeneracy never exceeds two. It can be proven that in one dimension, there are no degenerate bound states for normalizable wave functions. A sufficient condition on a piecewise continuous potential V and the energy E is the existence of two real numbers M , x 0 with M ≠ ≠ 0 such that ∀ ∀ x > x 0 we have V ( x ) − − E ≥ ≥ M 2 . In particular, V is bounded below in this criterion.