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In the present note, we uncover a remarkable connection between the length ofperiodic orbit of a classical particle enclosed in a class of 2-dimensionalplanar billiards and the energy of a quantum particle confined to move in anidentical region with infinitely high potential wall on the boundary. Weobserve that the quantum energy spectrum of the particle is in exact one-to-onecorrespondence with the spectrum of the amplitude squares of the periodicorbits of a classical particle for the class of integrable billiardsconsidered. We have established the results by geometric constructions andexploiting the method of reflective tiling and folding of classicaltrajectories. We have further extended the method to 3-dimensional billiardsfor which exact analytical results are scarcely available - exploiting thegeometric construction, we determine the exact energy spectra of two newtetrahedral domains which we believe are integrable. We test the veracity ofour results by comparing them with numerical results.

From Classical Periodic Orbits in Integrable $\pi$-Rational Billiards to Quantum Energy Spectrum