Spectral problem for the Schrodinger equation with singular potential polynomial of even degree

1996 ◽  
Vol 39 (5) ◽  
pp. 442-456
Author(s):  
A. S. Vshivteev ◽  
N. V. Norin ◽  
V. N. Sorokin
Keyword(s):  
Schrödinger Equation ◽  
Spectral Problem ◽  
Schrodinger Equation ◽  
Singular Potential

scholarly journals Evolution by Schrödinger equation of Aharonov–Berry superoscillations in centrifugal potential

2019 ◽  
Vol 475 (2225) ◽  
pp. 20180390 ◽  
Author(s):  
F. Colombo ◽  
J. Gantner ◽  
D. C. Struppa
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Singular Potential ◽  
Singular Potentials ◽  
Limit Solution ◽  
Quadratic Hamiltonian

In recent years, we have investigated the evolution of superoscillations under Schrödinger equation with non-singular potentials. In all those cases, we have shown that superoscillations persist in time. In this paper, we investigate the centrifugal potential, which is a singular potential, and we show that the techniques developed to study the evolution of superoscillations in the case of the Schrödinger equation with a quadratic Hamiltonian apply to this setting. We also specify, in the case of the centrifugal potential, the notion of super-shift of the limit solution, a fact explained in the last section of this paper. It then becomes apparent that superoscillations are just a particular case of super-shift.

Darboux Transformation for Coupled Non-Linear Schrödinger Equation and Its Breather Solutions

2017 ◽  
Vol 72 (1) ◽  
pp. 9-15 ◽  
Author(s):  
Lili Feng ◽  
Fajun Yu ◽  
Li Li
Keyword(s):  
Schrödinger Equation ◽  
Spectral Problem ◽  
Darboux Transformation ◽  
Schrodinger Equation ◽  
Soliton Solutions ◽  
Soliton Equations ◽  
Spectral Problems ◽  
Non Linear ◽  
And Mathematics

AbstractStarting from a 3×3 spectral problem, a Darboux transformation (DT) method for coupled Schrödinger (CNLS) equation is constructed, which is more complex than 2×2 spectral problems. A scheme of soliton solutions of an integrable CNLS system is realised by using DT. Then, we obtain the breather solutions for the integrable CNLS system. The method is also appropriate for more non-linear soliton equations in physics and mathematics.

PERTURBED SCHRÖDINGER EQUATION WITH SINGULAR POTENTIAL AND INITIAL DATA

2006 ◽  
Vol 08 (04) ◽  
pp. 433-452 ◽  
Author(s):  
MIRJANA STOJANOVIĆ
Keyword(s):  
Initial Data ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Singular Potential ◽  
Singular Integral Equations ◽  
Uniqueness Result ◽  
Stability Estimates ◽  
Delta Distribution ◽  
The Green Function ◽  
Regularized Derivatives

We consider linear Schrödinger equation perturbed by delta distribution with singular potential and the initial data. Due to the singularities appearing in the equation, we introduce two kinds of approximations: the parameter's approximation for potential and the initial data given by mollifiers of different growth and the approximation for the Green function for Schrödinger equation with regularized derivatives. These approximations reduce the perturbed Schrödinger equation to the family of singular integral equations. We prove the existence-uniqueness theorems in Colombeau space [Formula: see text], 1 ≤ p,q ≤ ∞, employing novel stability estimates (w.r.) to singular perturbations for ε → 0, which imply the statements in the framework of Colombeau generalized functions. In particular, we prove the existence-uniqueness result in [Formula: see text] and [Formula: see text] algebra of Colombeau.

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