Self-Similarity Among Energy Eigenstates

In a quantum system, different energy eigenstates have different properties or features, allowing us define a classifier to divide them into different groups. We find that the ratio of each type of energy eigenstates in an energy shell $[E_c-\mathrm{\Delta }E\mathrm{/}2,E_c+\mathrm{\Delta }E\mathrm{/}2]$ is invariant with changing width $\mathrm{\Delta }E$ or Planck constant $\mathrm{\hslash }$ as long as the number of eigenstates in the shell is statistically large enough. We give an argument that such self-similarity in energy eigenstates is a general feature for all quantum systems, which is further illustrated numerically with various quantum systems, including circular billiard, double top model, kicked rotor, and Heisenberg XXZ model.