For a 1D quantum particle with jump-discontinuous mass and scale-free boundary conditions, the eigenfunctions show erratic energy dependence and the system admits infinitely many distinct semiclassical limits indexed by points on a spectral curve in the two-torus.
Anantharaman,Quantum Ergodicity and Delocalization of Schr ¨odinger Eigenfunctions, Zurich Lectures in Advanced Mathematics, EMS Press, 2022
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Quantum Systems with jump-discontinuous mass. I
For a 1D quantum particle with jump-discontinuous mass and scale-free boundary conditions, the eigenfunctions show erratic energy dependence and the system admits infinitely many distinct semiclassical limits indexed by points on a spectral curve in the two-torus.