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 \([E_{c}-\Delta E/2,E_{c}+\Delta E/2]\) is invariant with changing width \(\Delta E\) or Planck constant \(\hbar\) 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.