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Stationary solutions of the Schr?dinger-Newton model - An ODE approach  [PDF]
Philippe Choquard,Joachim Stubbe,Marc Vuffray
Physics , 2007,
Abstract: We prove the existence and uniqueness of stationary spherically symmetric positive solutions for the Schr\"{o}dinger-Newton model in any space dimension $d$. Our result is based on an analysis of the corresponding system of second order differential equations. It turns out that $d=6$ is critical for the existence of finite energy solutions and the equations for positive spherically symmetric solutions reduce to a Lane-Emden equation for all $d\geq 6$. Our result implies in particular the existence of stationary solutions for two-dimensional self-gravitating particles and closes the gap between the variational proofs in $d=1$ and $d=3$.
The one-dimensional Schr?dinger-Newton equations  [PDF]
Ph. Choquard,J. Stubbe
Physics , 2007, DOI: 10.1007/s11005-007-0174-y
Abstract: We prove an existence and uniqueness result for ground states of one-dimensional Schr\"{o}dinger-Newton equations.
Bound states of two-dimensional Schr?dinger-Newton equations  [PDF]
Joachim Stubbe
Physics , 2008,
Abstract: We prove an existence and uniqueness result for ground states and for purely angular excitations of two-dimensional Schr\"{o}dinger-Newton equations. From the minimization problem for ground states we obtain a sharp version of a logarithmic Hardy-Littlewood-Sobolev type inequality.
Variational approximations for travelling solitons in a discrete nonlinear Schr?dinger equation  [PDF]
M. Syafwan,H. Susanto,S. M. Cox,B. A. Malomed
Physics , 2012, DOI: 10.1088/1751-8113/45/7/075207
Abstract: Travelling solitary waves in the one-dimensional discrete nonlinear Schr\"{o}dinger equation (DNLSE) with saturable onsite nonlinearity are studied. A variational approximation (VA) for the solitary waves is derived in an analytical form. The stability is also studied by means of the VA, demonstrating that the solitons are stable, which is consistent with previously published results. Then, the VA is applied to predict parameters of travelling solitons with non-oscillatory tails (\textit{embedded solitons}, ESs). Two-soliton bound states are considered too. The separation distance between the solitons forming the bound state is derived by means of the VA. A numerical scheme based on the discretization of the equation in the moving coordinate frame is derived and implemented using the Newton--Raphson method. In general, a good agreement between the analytical and numerical results is obtained. In particular, we demonstrate the relevance of the analytical prediction of characteristics of the embedded solitons.
Ground state solutions for nonlinear fractional Schr?dinger equations in $\mathbb{R}^N$  [PDF]
Simone Secchi
Mathematics , 2012, DOI: 10.1063/1.4793990
Abstract: We construct solutions to a class of Schr\"{o}dinger equations involving the fractional laplacian. Our approach is variational in nature, and based on minimization on the Nehari manifold.
Classical variational basis for Schr dinger quantum theory  [cached]
James G. Gilson
International Journal of Stochastic Analysis , 1988, DOI: 10.1155/s1048953388000218
Abstract: A two-dimensional classical hyperspace fluid theory of the vacuum which is an alternative to Schr dinger quantum theory involves a stirring energy density. This energy density, like entropy density, measures the local disturbance involved by the time evolution of a state of the system. The stirring energy density is shown to be the local source term for the rate of generation of action density with time. A classical fluid action principle is solved using the action of this source term without assuming quantum time evolution. Schr dinger quantum theory is recovered from this new action principle which is then discussed. Finally, the quantity made stationary by the variational principle measures disturbance in the environment as an angular change in the orientation of the whole system.
Bounds states of the Schr?dinger-Newton model in low dimensions  [PDF]
Joachim Stubbe,Marc Vuffray
Physics , 2008,
Abstract: We prove the existence of quasi-stationary symmetric solutions with exactly n>=0 zeros and uniqueness for n=0 for the Schr\"odinger-Newton model in one dimension and in two dimensions along with an angular momentum m>=0. Our result is based on an analysis of the corresponding system of second-order differential equations.
Schr?dinger-Newton "collapse" of the wave function  [PDF]
J. R. van Meter
Physics , 2011, DOI: 10.1088/0264-9381/28/21/215013
Abstract: It has been suggested that the nonlinear Schr\"odinger-Newton equation might approximate the coupling of quantum mechanics with gravitation, particularly in the context of the M{\o}ller-Rosenfeld semiclassical theory. Numerical results for the spherically symmetric, time-dependent, single-particle case are presented, clarifying and extending previous work on the subject. It is found that, for a particle mass greater than $\sim 1.14(\hbar^2/(G\sigma))^{1/3}$, a wave packet of width $\sigma$ partially "collapses" to a groundstate solution found by Moroz, Penrose, and Tod, with excess probability dispersing away. However, for a mass less than $\sim 1.14(\hbar^2/(G\sigma))^{1/3}$, the entire wave packet appears to spread like a free particle, albeit more slowly. It is argued that, on some scales (lower than the Planck scale), this theory predicts significant deviation from conventional (linear) quantum mechanics. However, owing to the difficulty of controlling quantum coherence on the one hand, and the weakness of gravity on the other, definitive experimental falsification poses a technologically formidable challenge.
Lie point symmetries and the geodesic approximation for the Schr?dinger-Newton equations  [PDF]
Oliver Robertshaw,Paul Tod
Mathematics , 2005, DOI: 10.1088/0951-7715/19/7/002
Abstract: We consider two problems arising in the study of the Schr\"odinger-Newton equations. The first is to find their Lie point symmetries. The second, as an application of the first, is to investigate an approximate solution corresponding to widely separated lumps of probability. The lumps are found to move like point particles under a mutual inverse-square law of attraction.
On the Schr{?}dinger-Newton equation and its symmetries: a geometric view  [PDF]
C. Duval,Serge Lazzarini
Physics , 2015, DOI: 10.1088/0264-9381/32/17/175006
Abstract: The \SN (SN) equation is recast on purely geometrical grounds, namely in terms of Bargmann structures over $(\d+1)$-dimensional Newton-Cartan (NC) spacetimes. Its maximal group of invariance, which we call the SN group, is determined as the group of conformal Bargmann automorphisms that preserve the coupled Schr\"odinger and NC gravitational field equations. Canonical unitary representations of the SN group are worked out, helping us recover, in particular, a very specific occurrence of dilations with dynamical exponent $z=(\d+2)/3$.
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