Bifurcation of nonlinear bound states in the periodic Gross-Pitaevskii equation with 𝒫𝒯-symmetry

2019 ◽  
Vol 150 (1) ◽  
pp. 171-204
Author(s):  
Tomáš Dohnal ◽  
Dmitry Pelinovsky
Keyword(s):  
Bound States ◽  
Periodic Potential ◽  
Sufficient Conditions ◽  
Amplitude Equation ◽  
One Dimension ◽  
Real Interval ◽  
Pitaevskii Equation ◽  
Spectral Bands ◽  
Cubic Nonlinear Schrödinger Equation ◽  
Small Imaginary Part

AbstractThe stationary Gross–Pitaevskii equation in one dimension is considered with a complex periodic potential satisfying the conditions of the 𝒫𝒯 (parity-time reversal) symmetry. Under rather general assumptions on the potentials, we prove bifurcations of 𝒫𝒯-symmetric nonlinear bound states from the end points of a real interval in the spectrum of the non-selfadjoint linear Schrödinger operator with a complex 𝒫𝒯-symmetric periodic potential. The nonlinear bound states are approximated by the effective amplitude equation, which bears the form of the cubic nonlinear Schrödinger equation. In addition, we provide sufficient conditions for the appearance of complex spectral bands when the complex 𝒫𝒯-symmetric potential has an asymptotically small imaginary part.

scholarly journals Universality of excited three-body bound states in one dimension

2021 ◽  
Author(s):  
Lucas Happ ◽  
Matthias Zimmermann ◽  
Maxim A Efremov
Keyword(s):  
Excited States ◽  
Bound States ◽  
Space Dimension ◽  
Binding Energies ◽  
Wave Functions ◽  
One Dimension ◽  
Strong Indication ◽  
Body System ◽  
Weakly Bound ◽  
Three Body

Abstract We study a heavy-heavy-light three-body system confined to one space dimension in the regime where an excited state in the heavy-light subsystems becomes weakly bound. The associated two-body system is characterized by (i) the structure of the weakly-bound excited heavy-light state and (ii) the presence of deeply-bound heavy-light states. The consequences of these aspects for the behavior of the three-body system are analyzed. We find a strong indication for universal behavior of both three-body binding energies and wave functions for different weakly-bound excited states in the heavy-light subsystems.

Bound States in One Dimension

1994 ◽  
pp. 22-44
Author(s):  
Siegmund Brandt ◽  
Hans Dieter Dahmen
Keyword(s):  
Bound States ◽  
One Dimension

Bound States in One Dimension

1994 ◽  
pp. 22-43
Author(s):  
Siegmund Brandt ◽  
Hans Dieter Dahmen
Keyword(s):  
Bound States ◽  
One Dimension

Bound States in One Dimension

1989 ◽  
pp. 20-39
Author(s):  
Siegmund Brandt ◽  
Hans Dieter Dahmen
Keyword(s):  
Bound States ◽  
One Dimension

Bound States in One Dimension

1992 ◽  
pp. 20-39
Author(s):  
Siegmund Brandt ◽  
Hans Dieter Dahmen
Keyword(s):  
Bound States ◽  
One Dimension

scholarly journals Derivation of the Time Dependent Gross–Pitaevskii Equation in Two Dimensions

2019 ◽  
Vol 372 (1) ◽  
pp. 1-69 ◽  
Author(s):  
Maximilian Jeblick ◽  
Nikolai Leopold ◽  
Peter Pickl
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Particle System ◽  
Nonlinear Schrodinger Equation ◽  
Two Dimensions ◽  
Pitaevskii Equation ◽  
Nonlinear Schrödinger ◽  
Cubic Nonlinear Schrödinger Equation ◽  
Reduced Density

Abstract We present microscopic derivations of the defocusing two-dimensional cubic nonlinear Schrödinger equation and the Gross–Pitaevskii equation starting from an interacting N-particle system of bosons. We consider the interaction potential to be given either by $$W_\beta (x)=N^{-1+2 \beta }W(N^\beta x)$$Wβ(x)=N-1+2βW(Nβx), for any $$\beta >0$$β>0, or to be given by $$V_N(x)=e^{2N} V(e^N x)$$VN(x)=e2NV(eNx), for some spherical symmetric, nonnegative and compactly supported $$W,V \in L^\infty ({\mathbb {R}}^2,{\mathbb {R}})$$W,V∈L∞(R2,R). In both cases we prove the convergence of the reduced density corresponding to the exact time evolution to the projector onto the solution of the corresponding nonlinear Schrödinger equation in trace norm. For the latter potential $$V_N$$VN we show that it is crucial to take the microscopic structure of the condensate into account in order to obtain the correct dynamics.

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