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Spectral statistics of quantized rational triangles reveal intermediate behavior, blending features of chaotic and integrable systems. This study analyzes large datasets to accurately characterize these unique quantum properties.

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Area of Science:

  • Quantum Chaos
  • Mathematical Physics
  • Billiard Systems

Background:

  • Rational triangles are simple pseudo-integrable systems with complex classical and quantum properties.
  • Understanding spectral statistics is key to classifying quantum system behavior.

Purpose of the Study:

  • To extensively study the spectral statistics of eight quantized rational triangles.
  • To accurately determine the type of spectral statistics exhibited by these systems.
  • To compare numerical findings with theoretical models.

Main Methods:

  • Numerical simulations of quantized rational triangles, including right-angled Veech and obtuse triangles.
  • Calculation of large spectral samples (up to one million energy levels) for each triangle.
  • Analysis of short-range (level spacing distributions) and long-range (number variance, spectral form factors) statistics.

Main Results:

  • Spectral statistics are of an intermediate type, exhibiting both level repulsion (chaotic) and exponential tails (integrable).
  • A finite value of level compressibility, characteristic of intermediate statistics, was observed.
  • Numerical data showed excellent agreement with the gamma distribution model.

Conclusions:

  • Quantized rational triangles display a unique intermediate spectral statistics.
  • The gamma distribution model accurately describes the observed spectral properties.
  • These findings contribute to the understanding of quantum chaos and pseudo-integrable systems.