scholarly journals Propagators of Generalized Schrödinger Equations Related by First-order Supersymmetry

10.14311/1366 ◽  
2011 ◽  
Vol 51 (2) ◽  
Author(s):  
A. Schulze-Halberg
Keyword(s):  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
First Order ◽  
Explicit Relation

We construct an explicit relation between propagators of generalized Schrödinger equations that are linked by a first-order supersymmetric transformation. Our findings extend and complement recent results on the conventional case [1].

scholarly journals Schrödinger Equations in Electromagnetic Fields: Symmetries and Noncommutative Integration

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1527
Author(s):  
Alexey Anatolievich Magazev ◽  
Maria Nikolaevna Boldyreva
Keyword(s):  
Schrödinger Equation ◽  
Electromagnetic Fields ◽  
Schrodinger Equation ◽  
Symmetry Algebra ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Exact Integration ◽  
First Order ◽  
Symmetry Properties ◽  
Noncommutative Integration

We study symmetry properties and the possibility of exact integration of the time-independent Schrödinger equation in an external electromagnetic field. We present an algorithm for constructing the first-order symmetry algebra and describe its structure in terms of Lie algebra central extensions. Based on the well-known classification of the subalgebras of the algebra e(3), we classify all electromagnetic fields for which the corresponding time-independent Schrödinger equations admit first-order symmetry algebras. Moreover, we select the integrable cases, and for physically interesting electromagnetic fields, we reduced the original Schrödinger equation to an ordinary differential equation using the noncommutative integration method developed by Shapovalov and Shirokov.

DARBOUX TRANSFORMATIONS FOR EFFECTIVE MASS SCHRÖDINGER EQUATIONS WITH ENERGY-DEPENDENT POTENTIALS

2008 ◽  
Vol 23 (03n04) ◽  
pp. 537-546 ◽  
Author(s):  
AXEL SCHULZE-HALBERG
Keyword(s):  
Effective Mass ◽  
Darboux Transformation ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Constant Mass ◽  
Darboux Transformations ◽  
First Order ◽  
Dependent Potential ◽  
Energy Dependent

We construct first-order Darboux transformations for stationary Schrödinger equations with position-dependent (effective) mass and energy-dependent potential. We point out characteristic differences to the corresponding Darboux transformation for constant mass.14

scholarly journals The inverse problem of two-state quantum systems with non-adiabatic static linear coupling

2020 ◽  
pp. 2050002
Author(s):  
Andrii Khrabustovskyi ◽  
Imen Rassas ◽  
Éric Soccorsi
Keyword(s):  
Inverse Problem ◽  
Quantum Systems ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Coupling Coefficients ◽  
Lipschitz Stability ◽  
Linear Coupling ◽  
First Order ◽  
Coupling Terms ◽  
Schrödinger System

We consider the inverse problem of determining the coupling coefficients in a two-state Schrödinger system. We prove a Lipschitz stability inequality for the zeroth- and first-order coupling terms by finitely many partial lateral measurements of the solution to the coupled Schrödinger equations.

scholarly journals Time Decay for Nonlinear Dissipative Schrödinger Equations in Optical Fields

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Nakao Hayashi ◽  
Chunhua Li ◽  
Pavel I. Naumkin
Keyword(s):  
Lower Bound ◽  
Positive Root ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Time Decay ◽  
Large Initial Data ◽  
Initial Value ◽  
Optical Fields ◽  
Gauge Invariant ◽  
Dissipative Condition

We consider the initial value problem for the nonlinear dissipative Schrödinger equations with a gauge invariant nonlinearityλup-1uof orderpn<p≤1+2/nfor arbitrarily large initial data, where the lower boundpnis a positive root ofn+2p2-6p-n=0forn≥2andp1=1+2forn=1.Our purpose is to extend the previous results for higher space dimensions concerningL2-time decay and to improve the lower bound ofpunder the same dissipative condition onλ∈C:Im⁡ λ<0andIm⁡ λ>p-1/2pRe λas in the previous works.

scholarly journals Infinitely many positive solutions of nonlinear Schrödinger equations

2021 ◽  
Vol 60 (2) ◽  
Author(s):  
Riccardo Molle ◽  
Donato Passaseo
Keyword(s):  
Positive Solutions ◽  
Radial Symmetry ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations

AbstractThe paper deals with the equation $$-\Delta u+a(x) u =|u|^{p-1}u $$ - Δ u + a ( x ) u = | u | p - 1 u , $$u \in H^1({\mathbb {R}}^N)$$ u ∈ H 1 ( R N ) , with $$N\ge 2$$ N ≥ 2 , $$p> 1,\ p< {N+2\over N-2}$$ p > 1 , p < N + 2 N - 2 if $$N\ge 3$$ N ≥ 3 , $$a\in L^{N/2}_{loc}({\mathbb {R}}^N)$$ a ∈ L loc N / 2 ( R N ) , $$\inf a> 0$$ inf a > 0 , $$\lim _{|x| \rightarrow \infty } a(x)= a_\infty $$ lim | x | → ∞ a ( x ) = a ∞ . Assuming that the potential a(x) satisfies $$\lim _{|x| \rightarrow \infty }[a(x)-a_\infty ] e^{\eta |x|}= \infty \ \ \forall \eta > 0$$ lim | x | → ∞ [ a ( x ) - a ∞ ] e η | x | = ∞ ∀ η > 0 , $$ \lim _{\rho \rightarrow \infty } \sup \left\{ a(\rho \theta _1) - a(\rho \theta _2) \ :\ \theta _1, \theta _2 \in {\mathbb {R}}^N,\ |\theta _1|= |\theta _2|=1 \right\} e^{\tilde{\eta }\rho } = 0 \quad \text{ for } \text{ some } \ \tilde{\eta }> 0$$ lim ρ → ∞ sup a ( ρ θ 1 ) - a ( ρ θ 2 ) : θ 1 , θ 2 ∈ R N , | θ 1 | = | θ 2 | = 1 e η ~ ρ = 0 for some η ~ > 0 and other technical conditions, but not requiring any symmetry, the existence of infinitely many positive multi-bump solutions is proved. This result considerably improves those of previous papers because we do not require that a(x) has radial symmetry, or that $$N=2$$ N = 2 , or that $$|a(x)-a_\infty |$$ | a ( x ) - a ∞ | is uniformly small in $${\mathbb {R}}^N$$ R N , etc. ....

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