On the Relationship between p-Analytic Functions and the Schrödinger Equation

2005 ◽  
pp. 487-496 ◽  
Author(s):  
Vladislav Kravchenko
Keyword(s):  
Schrödinger Equation ◽  
Analytic Functions ◽  
Schrodinger Equation ◽  
The Relationship

Painlevé analysis of the damped, driven nonlinear Schrödinger equation

1988 ◽  
Vol 109 (1-2) ◽  
pp. 109-126 ◽  
Author(s):  
Peter A. Clarkson
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Analytic Functions ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Completely Integrable ◽  
Real Analytic Functions ◽  
Image Position ◽  
Special Case

SynopsisIn this paper we apply the Painlevé tests to the damped, driven nonlinear Schrödinger equationwhere a(x, t) and b(x, t) are analytic functions of x and t, todetermine under what conditions the equation might be completely integrable. It is shown that (0.1) can pass the Painlevé tests only ifwhere α0(t),α1(t) and β(t) are arbitrary, real analytic functions of time. Furthermore, it is shown that in this special case, (0.1) may be transformed into the original nonlinear Schrödinger equation, which is known to be completely integrable.

scholarly journals On PT Symmetry Systems: Invariance, Conservation Laws, and Reductions

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
P. Masemola ◽  
A. H. Kara
Keyword(s):  
Schrödinger Equation ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Schrodinger Equation ◽  
Lie Symmetries ◽  
Pt Symmetry ◽  
Noether Symmetries ◽  
Cubic Schrödinger Equation ◽  
The Relationship

An analysis of a PT symmetric coupler with “gain in one waveguide and loss in another” is made; a transformation in the PT system and some assumptions results in a scalar cubic Schrödinger equation. We investigate the relationship between the conservation laws and Lie symmetries and investigate a Lagrangian, corresponding Noether symmetries, conserved vectors, and exact solutions via “double reductions.”

The symmetry group of the quantum harmonic oscillator in an electric field

Open Physics ◽  
2008 ◽  
Vol 6 (3) ◽  
Author(s):  
Juan Lejarreta ◽  
Jose Cerveró
Keyword(s):  
Schrödinger Equation ◽  
Electric Field ◽  
Harmonic Oscillator ◽  
Schrodinger Equation ◽  
Group Structure ◽  
Time Dependent ◽  
Frequency Function ◽  
General Group ◽  
Dependent Potential ◽  
The Relationship

AbstractIn this paper we present two results. First, we derive the most general group of infinitesimal transformations for the Schrödinger Equation of the general time-dependent Harmonic Oscillator in an electric field. The infinitesimal generators and the commutation rules of this group are presented and the group structure is identified. From here it is easy to construct a set of unitary operators that transform the general Hamiltonian to a much simpler form. The relationship between squeezing and dynamical symmetries is also stressed. The second result concerns the application of these group transformations to obtain solutions of the Schrödinger equation in a time-dependent potential. These solutions are believed to be useful for describing particles confined in boxes with moving boundaries. The motion of the walls is indeed governed by the time-dependent frequency function. The applications of these results to non-rigid quantum dots and tunnelling through fluctuating barriers is also discussed, both in the presence and in the absence of a time-dependent electric field. The differences and similarities between both cases are pointed out.

scholarly journals SUPERSYMMETRIC APPROACH TO EXACTLY SOLVABLE SYSTEMS WITH POSITION-DEPENDENT EFFECTIVE MASSES

2002 ◽  
Vol 17 (31) ◽  
pp. 2057-2066 ◽  
Author(s):  
BEŞİ GÖNÜL ◽  
BÜLENT GÖNÜL ◽  
DİLEK TUTCU ◽  
OKAN ÖZER
Keyword(s):  
Schrödinger Equation ◽  
Effective Mass ◽  
Schrodinger Equation ◽  
Hamiltonian Operator ◽  
One Dimensional ◽  
Exact Solvability ◽  
Exactly Solvable ◽  
The One ◽  
Dependent Mass ◽  
The Relationship

We discuss the relationship between exact solvability of the Schrödinger equation with a position-dependent mass and the ordering ambiguity in the Hamiltonian operator within the framework of supersymmetric quantum mechanics. The one-dimensional Schrödinger equation, derived from the general form of the effective mass Hamiltonian, is solved exactly for a system with exponentially changing mass in the presence of a potential with similar behaviour, and the corresponding supersymmetric partner Hamiltonians are related to the effective-mass Hamiltonians proposed in the literature.

scholarly journals Exact WKB methods in SU(2) Nf = 1

2022 ◽  
Vol 2022 (1) ◽  
Author(s):  
Alba Grassi ◽  
Qianyu Hao ◽  
Andrew Neitzke
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Fredholm Determinant ◽  
Quantization Condition ◽  
Direct Computation ◽  
Borel Summation ◽  
Local Solutions ◽  
Bps States ◽  
The Relationship ◽  
Good Agreement

Abstract We study in detail the Schrödinger equation corresponding to the four dimensional SU(2) $$ \mathcal{N} $$ N = 2 SQCD theory with one flavour. We calculate the Voros symbols, or quantum periods, in four different ways: Borel summation of the WKB series, direct computation of Wronskians of exponentially decaying solutions, the TBA equations of Gaiotto-Moore-Neitzke/Gaiotto, and instanton counting. We make computations by all of these methods, finding good agreement. We also study the exact quantization condition for the spectrum, and we compute the Fredholm determinant of the inverse of the Schrödinger operator using the TS/ST correspondence and Zamolodchikov’s TBA, again finding good agreement. In addition, we explore two aspects of the relationship between singularities of the Borel transformed WKB series and BPS states: BPS states of the 4d theory are related to singularities in the Borel transformed WKB series for the quantum periods, and BPS states of a coupled 2d+4d system are related to singularities in the Borel transformed WKB series for local solutions of the Schrödinger equation.

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