Operator Spectrum: Eigenvalue Distributions

Operator Type

H = -d²/dx² + x²
λ_n = 2n+1, n=0,1,2,...

Eigenvalues of the quantum harmonic oscillator: λ_n = 2n+1 (odd positive integers). Uniformly spaced spectrum — energy levels of the quantum harmonic oscillator.

Parameters

Matrix size N:

N = 12

Potential V(x): Spectral gap (min spacing):

Level repulsion in GUE: probability of small spacing ~ Δλ (Wigner surmise). Compare with Poisson = e^{-Δλ} (uncorrelated).

Operator Spectrum — Eigenvalue Distributions

Operator Type

Parameters