Exactly Solvable Schr\"odinger equations with Singularities: A Systematic Approach to Solving Complexified Potentials (part1)

• URL: http://arxiv.org/abs/2301.04138v1
• Date: Tue, 10 Jan 2023 05:39:43 GMT
• Title: Exactly Solvable Schr\"odinger equations with Singularities: A Systematic Approach to Solving Complexified Potentials (part1)
• Authors: Jamal Benbourenane
• Abstract summary: Extending the argument of the potential to a complex number leads to solving exactly the Schr"odinger equation. This paper gives a new perspective on how to solve the second-order linear differential equation written in normal form.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: This paper gives a new perspective on how to solve the second-order linear differential equation written in normal form. Extending the argument of the potential to a complex number leads to solving exactly the Schr\"odinger equation when the potential is complex using the factorization method. This method leads to solving two Riccati nonlinear equations and by constructing the only possible superpotential, the factorization method gives the eigenvalues and eigenfunctions in closed form for potentials satisfying the shape invariance property. Extending the potential to the complex argument has led to discovering new exactly solvable ones. In this first part, the basic superpotentials are divided into different groups, each group contains the superpotentials that share common terms. All of the already known solvable real potentials will fall into this category and are derived as special cases. This set of exactly solvable complexified potentials has already uncovered some of the properties of quantum mechanics, like the tunneling effect through the forbidden region, happening with high probabilities between multiwells, bound states in the continuum (BIC), and other properties. These results have potential applications in all fields of sciences, from physics, chemistry, biology, etc., where the eigenvalue problem plays an important role.

Related papers

• 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)
• The Half Transform Ansatz: Quarkonium Dynamics in Quantum Phase Space [0.0]
  We present a method to cast the Schrodinger Equation into a hyper-geometric form which can be solved for the phase space wave function and its energy eigenvalues. We also analyze the behavior of these wave functions, which suggest a correlation between radial momentum and the upper limit of existence in charm-anticharm mesons.
  arXiv  Detail & Related papers  (2023-03-28T23:38:57Z)
• Continuum energy eigenstates via the factorization method [0.0]
  We generalize the factorization method to energy continuum eigenstates. A "single-shot factorization" is enabled by writing the superpotential in a form that includes the logarithmic derivative of a confluent hypergeometric function.
  arXiv  Detail & Related papers  (2023-02-20T23:52:58Z)
• Nonlocalization of singular potentials in quantum dynamics [2.969705152497174]
  We demonstrate the superiority of a first-principle nonlocal model -- Wigner function -- in treating singular potentials. The nonlocal nature of the Wigner equation is fully exploited to convert the singular potential into the Wigner kernel with weak or even no singularity. Numerically converged Wigner functions under all these singular potentials are obtained with an operator splitting spectral method.
  arXiv  Detail & Related papers  (2023-01-18T04:24:33Z)
• Curvature-informed multi-task learning for graph networks [56.155331323304]
  State-of-the-art graph neural networks attempt to predict multiple properties simultaneously. We investigate a potential explanation for this phenomenon: the curvature of each property's loss surface significantly varies, leading to inefficient learning.
  arXiv  Detail & Related papers  (2022-08-02T18:18:41Z)
• Growth of entanglement of generic states under dual-unitary dynamics [77.34726150561087]
  Dual-unitary circuits are a class of locally-interacting quantum many-body systems. In particular, they admit a class of solvable" initial states for which, in the thermodynamic limit, one can access the full non-equilibrium dynamics. We show that in this case the entanglement increment during a time step is sub-maximal for finite times, however, it approaches the maximal value in the infinite-time limit.
  arXiv  Detail & Related papers  (2022-07-29T18:20:09Z)
• Decimation technique for open quantum systems: a case study with driven-dissipative bosonic chains [62.997667081978825]
  Unavoidable coupling of quantum systems to external degrees of freedom leads to dissipative (non-unitary) dynamics. We introduce a method to deal with these systems based on the calculation of (dissipative) lattice Green's function. We illustrate the power of this method with several examples of driven-dissipative bosonic chains of increasing complexity.
  arXiv  Detail & Related papers  (2022-02-15T19:00:09Z)
• Message Passing Neural PDE Solvers [60.77761603258397]
  We build a neural message passing solver, replacing allally designed components in the graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.
  arXiv  Detail & Related papers  (2022-02-07T17:47:46Z)
• Deformed Explicitly Correlated Gaussians [58.720142291102135]
  Deformed correlated Gaussian basis functions are introduced and their matrix elements are calculated. These basis functions can be used to solve problems with nonspherical potentials.
  arXiv  Detail & Related papers  (2021-08-10T18:23:06Z)
• Exactly Solvable Sextic Potential Having Symmetric Triple-Well Structure [0.0]
  We introduce a family of sextic potentials that are exactly solvable. For the first time, a family of triple-well potentials with their whole energy spectrum and wavefunctions using supersymmetry method.
  arXiv  Detail & Related papers  (2020-08-15T22:19:25Z)
• On the complex behaviour of the density in composite quantum systems [62.997667081978825]
  We study how the probability of presence of a particle is distributed between the two parts of a composite fermionic system. We prove that it is a non-perturbative property and we find out a large/small coupling constant duality. Inspired by the proof of KAM theorem, we are able to deal with this problem by introducing a cut-off in energies that eliminates these small denominators.
  arXiv  Detail & Related papers  (2020-04-14T21:41:15Z)

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.