Semiclassical treatment of the Schrödinger equation with relativistic kinematics and singular potential

1983 ◽  
Vol 28 (9) ◽  
pp. 2291-2296 ◽  
Author(s):  
P. Cea ◽  
G. Nardulli ◽  
G. Paiano
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Singular Potential ◽  
Relativistic Kinematics

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.

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.

scholarly journals The spinless relativistic Yukawa problem

2014 ◽  
Vol 29 (31) ◽  
pp. 1450195 ◽  
Author(s):  
Wolfgang Lucha ◽  
Franz F. Schöberl
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Bound State ◽  
Relativistic Kinematics ◽  
Nonzero Spin

Noticing renewed or increasing interest in the possibility to describe semirelativistic bound states (of either spin-zero constituents or, upon confining oneself to spin-averaged features, constituents with nonzero spin) by means of the spinless Salpeter equation generalizing the Schrödinger equation toward incorporation of effects caused by relativistic kinematics, we revisit this problem for interactions between bound-state constituents of Yukawa shape, by recalling and applying several well-known tools enabling to constrain the resulting spectra.

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