Long-range correlations in quantum spectra

Starting From Berry-Tabor trace formula, Einstein-Brillouin-Keller (EBK) qu antization condition is adopted to derive action quantization conditions of periodic orbits in two dimensional uncoupled oscillators. The corresponden ce relations between quantum levels and periodic orbits are found. They ind icate that two levels' cotributions to recurrence function coherently inter fere with each other if the two levels correspond to periodic orbits with i dentical topology. The peaks in recurrence spectra results from coherent in terference among levels whose corresponding periodic orbits have identical topology. There are long-range correlations among these levels.