A Nonlinear Schrödinger Model for Many-Particle Quantum Systems

Considering both effects of the s-wave scattering and the atom-atom interaction rather than only the effect of the s-wave scattering, we establish a nonlinear Schrödinger model for many-particle quantum systems and we prove the global existence of a solution to the model and obtain the expression of the solution. Furthermore, we show that the Hamilton energy and the total particle number both are conservative quantities.


Introduction
As the fundamental equations in quantum mechanics, nonlinear Schr ödinger NLS equations are usually chosen as models to describe many different kinds of many-particle quantum systems.One of the most typical examples is the Gross-Pitaevskii GP equation 1, 2 which was derived from a mean field approximation and used to describe the Bose-Einstein condensates BEC 3-7 for trapped dilute-gas at zero temperature.GP-equation has been extensively studied during the past decades 8-12 .Numerical analyses have been applied to GP-equation to investigate the dynamics of BEC 8, 9 ; analytical methods have also been used for solving the GP-equation to predict properties of the condensates 10 , and rigorous mathematical proof has been provided to guarantee the existence of travelling wave solutions to the GP-equation 11, 12 .Apart from these studies, work related with utilizing the generalized GP-equation to investigate physical phenomena still attracts much attention 13-16 .Coste 13 derived

The NLS Model for Many-Particle Quantum Systems
In order to obtain the NLS model, we take two steps.At first, we derive the new Hamilton energy from the correction of the original one.And then the NLS model can be obtained by applying the theory of quantum physics to the Hamiltonian.
At first, we recall the original GP-equation as follows: where m the boson mass, Planck constant, V x the external trapping potential, |ψ| 2 the density of bosons, and U the interaction "strength", given by where a is the s-wave scattering length with following physical interpretation: a > 0, the repulsive interaction, < 0, the attractive interaction.

2.3
Also, we note that the corresponding Hamilton energy of 2.1 as follows: where the first term in 2.4 , that is, represents the kinetic energy of bosons, the second term in 2.4 , that is, represents the external trapping potential energy and the last term in 2.4 , that is, represents the bosons interaction potential energy.Since the physical interpretation of the kinetic energy 2.5 and the external trapping potential energy 2.6 of GP-equation are the same as those of the NLS model and the only difference between them is the interaction potential energy, the Hamilton energy of the NLS model can be expressed as follows: where function g • : R → R is a C 1 function undetermined.Some essential information of function g can be derived from the following analysis.
For the NLS model, the interaction potential energy is built on the basis of the waveparticle duality in quantum mechanics, the assumptions that the atom-atom interaction be chosen to be the van der Waals force 31, 32 , and the interaction potential energy is changed continuously with density of particles.
The wave-particle duality in quantum mechanics points out that the many-particle quantum system has both the properties of wave and particle, which implies that the interactions among particles include both of the s-wave scattering and the atom-atom interaction.The interaction mechanisms of the former are the same as wave but the mechanisms of the latter are the same as particle.More precisely, if the scattering length is much smaller than the average distance between particles i.e., a d , the s-wave scattering takes the leading role, and if the scattering length is much bigger than the average distance between particles i.e., a d , the atom-atom interaction plays dominant role, or else the two both work.
If the scattering length is much smaller than the average distance between particles i.e., a d , the realistic interaction between particles can be approximated by the s-wave scattering as shown in 2.7 , then we can infer from 2.7 that g z and z follow the linear relationship, that is, g z c 0 z, 2.9 for general many-particle quantum systems provided that 0 ≤ z ≤ ρ 0 , where ρ 0 ∈ R is sufficiently small constant since the trapped gas is dilute and c 0 ∈ R or c 0 ∈ R − corresponds to repulsive and attractive case, respectively.If the scattering length is much bigger than the average distance between particles i.e., a d , the atom-atom interaction is chosen to be the van der Waals interaction, see 31, 32 and the references therein and the relationship between the atom-atom interaction potential energy and the average distance between particles can be depicted as shown in the Figure 1.
Since the average distance between particles and the density of particles are subject to conservation of mass, the relationship between them satisfies then together with Figure 1 the relationship between the atom-atom interaction potential energy and the density of particles can be derived as shown in the Figure 2.
Combining relationship 2.9 with Figure 2, we can get the effects of the s-wave scattering and the van der Waals interaction as shown in the Figure 3; 0, ρ 0 is the domain in which the s-wave scattering takes the leading role, ρ 1 , ρ 2 is the domain in which the van der Waals force plays the dominant role, and ρ 0 , ρ 1 is the domain in which the two both work.Considering both effects of the s-wave scattering and the van der Waals interaction and the assumption that the interaction potential energy is changed continuously with density of particles, we take function g as shown in the Figure 4.
Although the exact formula of g z can not be obtained from Figure 4, noting the fact that the density of particles in experiments never is infinite and bounded above, that is, the following conditions hold true: where c 1 is a positive constant and

2.13
where g 1 z , g 2 z are C 1 functions and g 2 satisfies where c 2 , c 3 , c 4 , c 5 are all positive constants.The physical conditions 2.12 and 2.13 can be written as the following mathematical form:

2.16
where g 1 z , g 2 z are C 1 functions and g 2 satisfies 2.17 According to the theory of quantum physics, that is, we obtain the NLS model as follows: where Ω be a bounded open set in R n n ≤ 3 and x, t ∈ Ω × 0, ∞ .Since |ψ| 2 < ρ 0 implies that the gas is dilute, the atom-atom interaction is effectively weak and the realistic interaction between particles can be approximated by the s-wave scattering.Under this condition and taking c 0 2πa 2 /m in 2.19 , the NLS model 2.19 yields the GP-equation, which implies that 2.19 can be regarded as a generalization of the GP-equation.

Mathematical Results of the NLS Model
For the NLS model 2.19 , we have the following global existence result.Theorem 3.1.Assume that g satisfies conditions 2.15 , 2.16 and 2.17 . Furthermore, for all x, t ∈ Ω × 0, ∞ , the solution can be expressed as follows: where

3.2
Proof.Firstly, we prove the global existence of solution to 2.19 in L ∞ 0, ∞ , H 1 Ω, C by Galerkin method.Let ψ u 1 iu 2 , by the theory of infinite dimensional Hamilton system proposed by Ma 24 , 2.19 be equivalent to the following equations: where

3.4
Let X 1 : Set X n , X n as follows:

3.5
Consider the ordinary equations as follows: Since 3.12 and 3.13 hold true, then Therefore there exists a subsequence, we still write it as By 2.15 , 2.16 , and 2.17 ; and Rellich-Kondrachov Compactness Theorem, we know operator 3.17 is weakly continuous.
Let n → ∞ in 3.8 , following equality: At last, we prove the global solution ψ u iv of 2.19 is in C 0 0, ∞ , L 2 Ω, C by semigroup theory.
Set operator Since the following equality: holds true for any u 1 , u 2 , ω 1 , ω 2 ∈ X 1 , then operator A : X 1 → X * 1 is an antisymmetric operator.By Stone Theorem, operator A is a generator of an unitary semigroup and the unitary semigroup can be represented as follows: T t cos tL − sin tL sin tL cos tL .

3.21
The solution of 3.3 can be represented as follows:
Furthermore, the solution ψ can be expressed as follows:

3.24
Remark 3.2.In 24 , it was proved that all dynamic equations in quantum physics, that is, can be equivalently expressed as the following Hamilton form: where H the Hamilton energy, ψ k the wave function and

The Conservation Laws
For the NLS model 2.19 , we prove that the Hamilton energy and the total particle number both are conservative quantities.
which imply that the energy H and the total particle number N both are conservative quantities of 2.19 .

Discussion
Taking the dense fluid of strongly interacting bosons as mixture of normal fluid and superfluid, and substituting the rule of Galilean invariance for local gauge invariance, Coste 13 made correction for the Lagrangian of the original GP-model by adding the degrees of freedom corresponding to the normal fluid.As a result, they obtained the generalized GPequation which can describe both the two-fluid model and the Hills-Roberts model under two different circumstances.
Building on the works of Gardiner 14 and Castin and Dum 15 , Gardiner and Morgan 16 derived the generalized GP-equation which can describe the condensate and noncondensate dynamics of Bose-Einstein dilute atomic gas at finite temperature.It is worthwhile to mention that unlike using expansion in inverse powers of the total number of particles in 14 , Gardiner and Morgan 16 used expansion in powers of ration of noncondensate to condensate particle numbers and constructed the approximate Hamiltonian of third-order.
As the same as the above works 13, 16 , the manuscript presented here also derive the generalized GP-equation from the correction of the original Hamiltonian.In addition, collapse blow up is prevented for our model and also for condensate and noncondensate dynamic model 16 when accounting for fluctuations.
But differences do exist.One of the differences is that the two above works are based on the classification of the state of particle or the state of fluid , that is, condensate and noncondensate or normal fluid and superfluid , but ours focus on the classification of interaction of particles, that is, the s-wave scattering and the atom-atom interaction.And another difference is the mechanism on which above related models build.The two-fluid model 13 respect Galilean invariance rule.The condensate and noncondensate dynamic model 16 obey the formalism of number-conserving.However, our model builds on the basis of the wave-particle duality in quantum mechanics.
It must be pointed out that our model is valid only on three assumptions.One is that the interactions among particles include only two effects: the s-wave scattering and the atom-atom interaction.Consequently, there is no other effect.Another one is that atom-atom interaction be chosen to be the van der Waals force.And the third one is that the interaction potential energy is changed continuously with density of particles.
Meanwhile, it is that the global solution to the NLS model is well defined and collapse blow-up is prevented guaranteeing the model is reasonable from the mathematical viewpoint, but further physical experiments should be done to verify this model.Especially, data from future experiments combining with the conditions 2.15 and 2.16 may give the exact expression of function g.
Inspired by the classification of the state of particle i.e., condensate and noncondensate in works 9, 10, 15, 16 , and taking into account of the classification of the interaction of particles i.e., the s-wave scattering and the atom-atom interaction , further investigation on the model of general many-particle quantum systems is necessary in future.

Conclusions
Building on the basis of the wave-particle duality in quantum mechanics and assumption that the interactions among particles include only two effects: the s-wave scattering and the atomatom interaction, we derive the new Hamilton from the correction of the original one.From the new Hamiltonian we derive the NLS model for more general many-particle quantum systems.Expression of the global solution to the NLS model is obtained.Collapse blow-up is prevented for the model.Conservation laws hold true both for the Hamilton energy and the total particle number.

Figure 1 :Figure 2 :
Figure 1:The atom-atom interaction potential energy as a function of the average distance between particles.

Figure 3 :Figure 4 :
Figure 3: Effects of the s-wave scattering and the van der Waals interaction.
Theorem 4.1.Assume that g satisfies conditions 2.15 , 2.16 and 2.17 , if V x ∈ L 2 Ω , then for any ψ 0 ∈ H 1 Ω, C , there exist two conservative quantities for 2.19 : the energy Proof.Since ψ u iv is the solution of 2.19 , then u, v is the solution of 3.3 and we have the following equalities: