Commentary on: [L 41] On the Uniqueness of the Potential in a Schrödinger Equation for a Given Asymptotic Phase [L 43] The Inverse Sturm-Liouville Problem [L 58] Certain Explicit Relationships between Phase Shift and Scattering Potential

1998 ◽  
pp. 159-163
Author(s):  
D. H. Sattinger
Keyword(s):  
Phase Shift ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Liouville Problem ◽  
Sturm Liouville ◽  
Scattering Potential ◽  
Asymptotic Phase

Nonadditive quantum mechanics as a Sturm–Liouville problem

2016 ◽  
Vol 27 (04) ◽  
pp. 1650047 ◽  
Author(s):  
João P. M. Braga ◽  
Raimundo N. Costa Filho
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Time Evolution ◽  
Schrodinger Equation ◽  
Liouville Problem ◽  
Hamiltonian Operator ◽  
Sturm Liouville

The modified Schrödinger equation obtained by Costa Filho et al. [Phys. Rev. A 84, 050102(R) (2011)] is shown to be a Sturm–Liouville problem. This demonstration guarantees that Hamiltonian eigenvalues obtained in this formalism are real. It also allows us to show that, regardless of the non-Hermitian characteristic of the Hamiltonian operator in the Hilbert space, its time evolution remains unitary.

Nonlinear Stage of Modulation Instability for a Fifth-Order Nonlinear Schrödinger Equation

2017 ◽  
Vol 72 (11) ◽  
pp. 1071-1075 ◽  
Author(s):  
Hui-Xian Jia ◽  
Dong-Ming Shan
Keyword(s):  
Phase Shift ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Modulation Instability ◽  
Rogue Wave ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Wave Solutions ◽  
Fifth Order

AbstractIn this article, a fifth-order nonlinear Schrödinger equation, which can be used to characterise the solitons in the optical fibre and inhomogeneous Heisenberg ferromagnetic spin system, has been investigated. Akhmediev breather, Kuzentsov soliton, and generalised soliton have all been attained via the Darbox transformation. Propagation and interaction for three-type breathers have been studied: the types of breather are determined by the module and complex angle of parameter ξ; interaction between Akhmediev breather and generalised soliton displays a phase shift, whereas the others do not. Modulation instability of the generalised solitons have been analysed: a small perturbation can develop into a rogue wave, which is consistent with the results of rogue wave solutions.

High-algebraic, high-phase lag methods for accurate computations for the elastic- scattering phase shift problem

1998 ◽  
Vol 76 (6) ◽  
pp. 473-493 ◽  
Author(s):  
T E Simos
Keyword(s):  
Elastic Scattering ◽  
Phase Shift ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Finite Difference Methods ◽  
Phase Lag ◽  
New Methods ◽  
Shift Problem ◽  
Algebraic Order ◽  
Scattering Phase

A family of three new hybrid eighth-algebraic-order two-step methods with phase lag of order 16, 18, and 20 are developed for computing elastic-scattering phase shifts of the one-dimensional Schrödinger equation. Based on these new methods, we obtain some new embedded variable-step procedures for the numerical integration of the Schrödinger equation. Numerical results obtained for both the integration of the phase-shift problem for the well known case of the Lennard–Jones potential and the integration of coupled differential equation arising from the Schrödinger equation show that these new methods are better than other finite-difference methods. PACS Nos.: 02.00, 02.70, 03.00, 03.65

AN EIGHTH-ORDER METHOD WITH MINIMAL PHASE-LAG FOR ACCURATE COMPUTATIONS FOR THE ELASTIC SCATTERING PHASE-SHIFT PROBLEM

1996 ◽  
Vol 07 (06) ◽  
pp. 825-835 ◽  
Author(s):  
T. E. SIMOS
Keyword(s):  
Elastic Scattering ◽  
Phase Shift ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Finite Difference Methods ◽  
New Method ◽  
Phase Lag ◽  
Step Method ◽  
Shift Problem ◽  
Scattering Phase

A new hybrid eighth-algebraic-order two-step method with phase-lag of order ten is developed for computing elastic scattering phase shifts of the one-dimensional Schrödinger equation. Based on this new method and on the method developed recently by Simos we obtain a new variable-step procedure for the numerical integration of the Schrödinger equation. Numerical results obtained for the integration of the phase shift problem for the well known case of the Lenard–Jones potential show that this new method is better than other finite difference methods.

A Sturm-Liouville Equation: The Time-Independent One-Dimensional Schrödinger Equation

2017 ◽  
Author(s):  
John A. Adam
Keyword(s):  
Schrödinger Equation ◽  
Liouville Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Classical Case ◽  
Liouville Theory ◽  
Weyl’S Theorem ◽  
One Dimensional ◽  
Sturm Liouville ◽  
The One

This chapter examines the mathematical properties of the time-independent one-dimensional Schrödinger equation as they relate to Sturm-Liouville problems. The regular Sturm-Liouville theory was generalized in 1908 by the German mathematician Hermann Weyl on a finite closed interval to second-order differential operators with singularities at the endpoints of the interval. Unlike the classical case, the spectrum may contain both a countable set of eigenvalues and a continuous part. The chapter first considers the one-dimensional Schrödinger equation in the standard dimensionless form (with independent variable x) and various relevant theorems, along with the proofs, before discussing bound states, taking into account bound-state theorems and complex eigenvalues. It also describes Weyl's theorem, given the Sturm-Liouville equation, and looks at two cases: the limit point and limit circle. Four examples are presented: an “eigensimple” equation, Bessel's equation of order ? greater than or equal to 0, Hermite's equation, and Legendre's equation.

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