Although the original proof by Valentine Bargmann is quite technical, the main idea follows from two general theorems on ordinary differential equations, the Sturm Oscillation Theorem and the Sturm-Picone Comparison Theorem. If we denote by u 0 ℓ ℓ the wave function subject to the given potential with total energy E = 0 and azimuthal quantum number ℓ ℓ , the Sturm Oscillation Theorem implies that N ℓ ℓ equals the number of nodes of u 0 ℓ ℓ . From the Sturm-Picone Comparison Theorem, it follows that when subject to a stronger potential W (i.e. W ( r ) ≤ ≤ V ( r ) for all r ∈ ∈ R 0 + ), the number of nodes either grows or remains the same. Thus, more specifically, we can replace the potential V by − − | V | . For the corresponding wave function with total energy E = 0 and azimuthal quantum number ℓ ℓ , denoted by ϕ ϕ 0 ℓ ℓ , the radial Schrödinger equation becomes