2023/11/10 by Alonso Contreras-Astorga, Contreras-Astorga, Alonso, A. M. Escobar-Ruiz +3 · 2 citations
Physics and Astronomy · #Quantum Mechanics and Non-Hermitian Physics #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2311.06230
In this paper, the SUSY partner Hamiltonians of the quasi-exactly solvable (QES) sextic potential V\rm qes(x) = ν x6 + 2 ν μ x4 + [μ2-(4N+3)ν] x2, N ∈ ℤ+, are revisited from a Lie algebraic perspective. It is demonstrated that, in the variable τ=x2, the underlying \mathfraksl2(ℝ) hidden algebra of V\rm qes(x) is inherited by its SUSY partner potential V1(x) only for N=0. At fixed N>0, the algebraic polynomial operator h(x, ∂x; N) that governs the N exact eigenpolynomial solutions of V1 is derived explicitly. These odd-parity solutions appear in the form of zero modes. The potential V1 can be represented as the sum of a polynomial and rational parts. In particular, it is shown that the polynomial component is given by V\rm qes with a different non-integer (cohomology) parameter N1=N-(3)/(2). A confluent second-order SUSY transformation is also implemented for a modified QES sextic potential possessing the energy reflection symmetry. By taking N as a continuous real constant and using the Lagrange-mesh method, highly accurate values (∼ 20 s. d.) of the energy En=En(N) in the interval N ∈ [-1,3] are calculated for the three lowest states n=0,1,2 of the system. The critical value Nc above which tunneling effects (instanton-like terms) can occur is obtained as well. At N=0, the non-algebraic sector of the spectrum of V\rm qes is described by means of compact physically relevant trial functions. These solutions allow us to determine the effects in accuracy when the first-order SUSY approach is applied on the level of approximate eigenfunctions.