scholarly journals Solutions of the time-independent Schrödinger equation by uniformization on the unit circle

2019 ◽  
Vol 18 (1) ◽  
pp. 157-165
Author(s):  
Kazimierz Rajchel
Keyword(s):  
Schrödinger Equation ◽  
Unit Circle ◽  
Riccati Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Logarithmic Derivative ◽  
Quantization Condition ◽  
State Function ◽  
Winding Number ◽  
Quantization Rule

Abstract The idea presented here of a general quantization rule for bound states is mainly based on the Riccati equation which is a result of the transformed, time-independent, one-dimensional Schrödinger equation. The condition imposed on the logarithmic derivative of the ground state function W0 allows to present the Riccati equation as the unit circle equation with winding number equal to one which, by appropriately chosen transformations, can be converted into the unit circle equation with multiple winding number. As a consequence, a completely new quantization condition, which gives exact results for any quantum number, is obtained.

The second Exton potential for the Schrödinger equation

2019 ◽  
Vol 34 (24) ◽  
pp. 1950195
Author(s):  
Artur M. Ishkhanyan ◽  
Jacek Karwowski
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Fractional Power ◽  
Fundamental Solutions ◽  
Transcendental Equation ◽  
Quantization Condition ◽  
Hermite Functions ◽  
Energy Quantization ◽  
Half Axis

Analytical solutions of the Schrödinger equation with a singular, fractional-power potential, referred to as the second Exton potential, are derived and analyzed. The potential is defined on the positive half-axis and supports an infinite number of bound states. It is conditionally integrable and belongs to a biconfluent Heun family. The fundamental solutions are expressed as irreducible linear combinations of two Hermite functions of a scaled and shifted argument. The energy quantization condition results from the boundary condition imposed at the origin. For the exact eigenvalues, which are solutions of a transcendental equation involving two Hermite functions, highly accurate approximation by simple closed-form expressions is derived. The potential is a good candidate for the description of quark–antiquark interaction.

scholarly journals EXACT QUANTIZATION RULES FOR BOUND STATES OF THE SCHRÖDINGER EQUATION

2005 ◽  
Vol 14 (04) ◽  
pp. 599-610 ◽  
Author(s):  
ZHONG-QI MA ◽  
BO-WEI XU
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Three Dimensional ◽  
Spherically Symmetric ◽  
Quantization Rule ◽  
One Dimensional ◽  
Exact Quantization Rule ◽  
The One ◽  
Quantization Rules

An exact quantization rule for the bound states of the one-dimensional Schrödinger equation is presented and is generalized to the three-dimensional Schrödinger equation with a spherically symmetric potential.

Solution of the Schrödinger equation for bound states in closed form

1982 ◽  
Vol 26 (1) ◽  
pp. 662-664 ◽  
Author(s):  
Edgardo Gerck ◽  
Jason A. C. Gallas ◽  
Augusto B. d'Oliveira
Keyword(s):  
Schrödinger Equation ◽  
Closed Form ◽  
Schrodinger Equation ◽  
Bound States

scholarly journals Observation of Floquet solitons in a topological bandgap

Science ◽  
2020 ◽  
Vol 368 (6493) ◽  
pp. 856-859 ◽  
Author(s):  
Sebabrata Mukherjee ◽  
Mikael C. Rechtsman
Keyword(s):  
Schrödinger Equation ◽  
Topological Insulator ◽  
Kerr Effect ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Optical Kerr Effect ◽  
Pitaevskii Equation ◽  
Waveguide Array ◽  
Winding Number ◽  
Interparticle Interactions

Topological protection is a universal phenomenon that applies to electronic, photonic, ultracold atomic, mechanical, and other systems. The vast majority of research in these systems has explored the linear domain, where interparticle interactions are negligible. We experimentally observed solitons—waves that propagate without changing shape as a result of nonlinearity—in a photonic Floquet topological insulator. These solitons exhibited distinct behavior in that they executed cyclotron-like orbits associated with the underlying topology. Specifically, we used a waveguide array with periodic variations along the waveguide axis, giving rise to nonzero winding number, and the nonlinearity arose from the optical Kerr effect. This result applies to a range of bosonic systems because it is described by the focusing nonlinear Schrödinger equation (equivalently, the attractive Gross-Pitaevskii equation).

A class of time periodic Hamiltonians with no bound states

1987 ◽  
Vol 105 (1) ◽  
pp. 37-42
Author(s):  
H. Kaneta
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
All Solutions ◽  
Time Periodic

SynopsisWe generalise the Paley–Wiener closedness theorem and apply it to a class of time periodic Hamiltonians to show that all solutions to the corresponding Schrodinger equation decay.

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