On symmetric Tetranacci polynomials in mathematics and physics. (English) Zbl 1533.81062
Summary: In this manuscript, we introduce (symmetric) Tetranacci polynomials \(\xi_j\) as a twofold generalization of ordinary Tetranacci numbers, considering both non unity coefficients and generic initial values. We derive a complete closed form expression for any \(\xi_j\) with the key feature of a decomposition in terms of generalized Fibonacci polynomials. For suitable conditions, \(\xi_j\) can be understood as the superposition of standing waves. The issue of Tetranacci polynomials originated from their application in condensed matter physics. We explicitly demonstrate the approach for the spectrum, eigenvectors, Green’s functions and transmission probability for an atomic tight binding chain exhibiting both nearest and next nearest neighbor processes. We demonstrate that in topological trivial models, complex wavevectors can form bulk states as a result of the open boundary conditions. We describe how effective next nearest neighbor bonding is engineered in state of the art theory/experiment exploiting onsite degrees of freedom and close range hopping. We argue about experimental tune ability and on-demand complex wavevectors.
MSC:
81R30 | Coherent states |
53D20 | Momentum maps; symplectic reduction |
81S10 | Geometry and quantization, symplectic methods |
35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
82B41 | Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics |
35K45 | Initial value problems for second-order parabolic systems |
35P15 | Estimates of eigenvalues in context of PDEs |
35J08 | Green’s functions for elliptic equations |
35G15 | Boundary value problems for linear higher-order PDEs |
Keywords:
Tetranacci polynomials; Fibonacci decomposition; complex wavevectors; transcendental momentum quantization; bulk-boundary correspondence; (engineering) next nearest neighbor hoppingReferences:
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