Quantum chaos, scattering, and BICs on smooth potentials

In this dissertation, quantum signatures of chaos and scattering dynamics on several smooth potential models are investigated. In the second chapter, we study chaos in a honeycomb lattice reminiscent of graphene [Phys. Rev. E 93, 012204 (2016)].We show classical chaos to exist in a wide energy range, identify key stable and unstable orbits, and find a quantum eigenstate with irregular nodal structure that’s suggestive of quantum chaos. Newer work for this dissertation shows this model can approximately reproduce the band structure of graphene with only two tuning parameters at low computational cost. In the third chapter, we study a square lattice of Gaussians with variable width [Phys. Rev. E 95, 052213 (2017)]. We find underlying classical chaos correlates with more avoided crossings in band structure, causing frequent mixing of eigenstates. We also find hints of eigenstate scarring. Newer work for this dissertation shows some similarity to the classic Sinai billiard, but with stronger chaos due to the smooth Gaussian peaks, and mixed dynamics due to the potential saddles. In the fourth chapter, we further study quantum chaos in a square lattice [Chaos 27, 104604 (2017)]. We give stronger evidence that underlying chaos causes avoided crossings in the band structure, both on and off the Brillouin zone symmetry lines, and give further examples of eigenstate mixing. We then provide stronger evidence of quantum chaos via energy level spacing distributions. In the fifth chapter, we study scattering from a chaotic triple Gaussian potential [Phys. Rev. E 97, 042206 (2018)]. We use Wigner-Eisenbud (W-E) reaction matrix scattering theory to find quasibound state resonances, and corroborate them with Wigner-Smith delay time plots, reaction region eigenstate plots, and the non-Hermitian Hamiltonian method. We also give evidence of classical and quantum chaos. In the sixth chapter, we use W-E scattering theory to demonstrate that quantum bound states called BICs and long-lived quasibound states exist in the scattering continuum of a quasi-1D Gaussian well lattice [Physica B 571, 15 (2019)]. The BICs come in two types: one protected from decaying by reflection symmetry, the other protected by discrete translational symmetry (conservation of Bloch momentum). The long-lived quasibound states come in odd-even pairs and exist to high energies. The reaction region eigenstates corresponding to these special scattering states are shown to be localized and largely independent of the reaction region boundary (often called the "channel radius").Physic