Supersymmetric Expansion Algorithm and complete analytical solution for the Hulthén and anharmonic potentials

• URL: http://arxiv.org/abs/2405.01367v1
• Date: Thu, 2 May 2024 15:13:06 GMT
• Title: Supersymmetric Expansion Algorithm and complete analytical solution for the Hulthén and anharmonic potentials
• Authors: M. Napsuciale, S. Rodríguez, M. Kirchbach,
• Abstract summary: An algorithm for providing analytical solutions to Schr"odinger's equation with non-exactly solvable potentials is elaborated. It represents a symbiosis between the logarithmic expansion method and the techniques of the superymmetric quantum mechanics.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: An algorithm for providing analytical solutions to Schr\"{o}dinger's equation with non-exactly solvable potentials is elaborated. It represents a symbiosis between the logarithmic expansion method and the techniques of the superymmetric quantum mechanics as extended toward non shape invariant potentials. The complete solution to a given Hamiltonian $H_{0}$ is obtained from the nodeless states of the Hamiltonian $H_{0}$ and of a set of supersymmetric partners $H_{1}, H_{2},..., H_{r}$. The nodeless states (dubbed "edge" states) are unique and in general can be ground or excited states. They are solved using the logarithmic expansion which yields an infinite systems of coupled first order hierarchical differential equations, converted later into algebraic equations with recurrence relations which can be solved order by order. We formulate the aforementioned scheme, termed to as "Supersymmetric Expansion Algorithm'' step by step and apply it to obtain for the first time the complete analytical solutions of the three dimensional Hulth\'en--, and the one-dimensional anharmonic oscillator potentials.

Related papers

• Solving the homogeneous Bethe-Salpeter equation with a quantum annealer [34.173566188833156]
  The homogeneous Bethe-Salpeter equation (hBSE) was solved for the first time by using a D-Wave quantum annealer. A broad numerical analysis of the proposed algorithms was carried out using both the proprietary simulated-anneaing package and the D-Wave Advantage 4.1 system.
  arXiv  Detail & Related papers  (2024-06-26T18:12:53Z)
• Prepotential Approach: a unified approach to exactly, quasi-exactly, and rationally extended solvable quantal systems [0.0]
  We give a brief overview of a simple and unified way, called the prepotential approach. It treats both exact and quasi-exact solvabilities of the one-dimensional Schr"odinger equation. We illustrate the approach by several paradigmatic examples of Hermitian and non-Hermitian Hamiltonians with real energies.
  arXiv  Detail & Related papers  (2023-10-22T11:40:00Z)
• Renormalization group and spectra of the generalized P\"oschl-Teller potential [0.0]
  We study the P"oschl-Teller potential $V(x) = alpha2 g_s sinh-2(alpha x) + alpha2 g_c cosh-2(alpha x)$, for every value of the dimensionless parameters $g_s$ and $g_c singularity. We show that supersymmetry of the potential, when present, is also spontaneously broken, along with conformal symmetry.
  arXiv  Detail & Related papers  (2023-08-08T21:44:55Z)
• A Newton-CG based barrier-augmented Lagrangian method for general nonconvex conic optimization [53.044526424637866]
  In this paper we consider finding an approximate second-order stationary point (SOSP) that minimizes a twice different subject general non conic optimization. In particular, we propose a Newton-CG based-augmentedconjugate method for finding an approximate SOSP.
  arXiv  Detail & Related papers  (2023-01-10T20:43:29Z)
• Towards Antisymmetric Neural Ansatz Separation [48.80300074254758]
  We study separations between two fundamental models of antisymmetric functions, that is, functions $f$ of the form $f(x_sigma(1), ldots, x_sigma(N)) These arise in the context of quantum chemistry, and are the basic modeling tool for wavefunctions of Fermionic systems.
  arXiv  Detail & Related papers  (2022-08-05T16:35:24Z)
• A unified approach to the nonlinear Rabi models [0.0]
  An analytical approach is proposed and applied to study the two-photon, two-mode and intensity-dependent Rabi models. This work paves a way for the analysis of the novel physics in nonlinear quantum optics.
  arXiv  Detail & Related papers  (2022-06-20T14:29:14Z)
• $O(N^2)$ Universal Antisymmetry in Fermionic Neural Networks [107.86545461433616]
  We propose permutation-equivariant architectures, on which a determinant Slater is applied to induce antisymmetry. FermiNet is proved to have universal approximation capability with a single determinant, namely, it suffices to represent any antisymmetric function. We substitute the Slater with a pairwise antisymmetry construction, which is easy to implement and can reduce the computational cost to $O(N2)$.
  arXiv  Detail & Related papers  (2022-05-26T07:44:54Z)
• The Franke-Gorini-Kossakowski-Lindblad-Sudarshan (FGKLS) Equation for Two-Dimensional Systems [62.997667081978825]
  Open quantum systems can obey the Franke-Gorini-Kossakowski-Lindblad-Sudarshan (FGKLS) equation. We exhaustively study the case of a Hilbert space dimension of $2$.
  arXiv  Detail & Related papers  (2022-04-16T07:03:54Z)
• Mean-Square Analysis with An Application to Optimal Dimension Dependence of Langevin Monte Carlo [60.785586069299356]
  This work provides a general framework for the non-asymotic analysis of sampling error in 2-Wasserstein distance. Our theoretical analysis is further validated by numerical experiments.
  arXiv  Detail & Related papers  (2021-09-08T18:00:05Z)
• Spectral Analysis of Product Formulas for Quantum Simulation [0.0]
  We show that the Trotter step size needed to estimate an energy eigenvalue within precision can be improved in scaling from $epsilon$ to $epsilon1/2$ for a large class of systems. Results partially generalize to diabatic processes, which remain in a narrow energy band separated from the rest of the spectrum by a gap.
  arXiv  Detail & Related papers  (2021-02-25T03:17:25Z)
• Bernstein-Greene-Kruskal approach for the quantum Vlasov equation [91.3755431537592]
  The one-dimensional stationary quantum Vlasov equation is analyzed using the energy as one of the dynamical variables. In the semiclassical case where quantum tunneling effects are small, an infinite series solution is developed.
  arXiv  Detail & Related papers  (2021-02-18T20:55:04Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.