The Grillakis-Machedon-Margetis Second order corrections to mean field evolution for weakly interacting Bosons in the case of 3-body interactions

TL;DR

This paper extends previous results on mean field evolution for weakly interacting bosons to include 3-body interactions, demonstrating that the core findings remain valid with minor modifications.

Contribution

It adapts existing mean field results to the 3-body interaction case, broadening the applicability of the original framework.

Findings

Results hold for 3-body interactions with minor modifications

Extends mean field evolution analysis to more complex interactions

Maintains validity of previous results in a new interaction setting

Abstract

This note shows that, with a little modification, the results in arXiv:0904.0158 hold in the 3-body interactions case.

Equations

$$i\frac{\partial \varphi }{\partial t}+\Delta \varphi +(v^{\prime }\ast |\varphi {|}^2)\varphi =0$$

$$(i{u}_t+u{g}^T+gu-(I+p)m)=(i{p}_t+[g,p]+u\overline{m})(1+p{)}^{-1}u,$$

$$u(t,x,y):=\sinh (k):=k+\frac{1}{3!}k\overline{k}k+\dots ,$$

$$\delta (x-y)+p(t,x,y):=\cosh (k):=\delta (x-y)+\frac{1}{2!}k\overline{k}+\dots ,$$

$$g(t,x,y):=-{\Delta }_x\delta (x-y)-v(x-y)\varphi (t,x)\overline{\varphi }(t,y)-(v\ast |\varphi {|}^2)(t,x)\delta (x-y),$$

$$m(t,x,y):=v(x-y)\overline{\varphi }(t,x)\overline{\varphi }(t,y).$$

$$B(t)=\frac{1}{2}\int \left(k(t,x,y)a_x{a}_y-\overline{k}(t,x,y)a_x^{\ast }{a}_y^{\ast }\right)dxdy$$

$$d(t,x,y)=$$

$$-$$

$$\Vert e^{-\sqrt{N}A(t)}{e}^{-B(t)}{e}^{-i{\int }_0^t(N{\chi }_0(s)+{\chi }_1(s))ds}\Omega -e^{it{H}_N^{\prime }}{e}^{-\sqrt{N}A(0)}\Omega {\Vert }_{\mathcal{F}}$$

$$\le \frac{{\int }_0^{t}f(s)ds}{\sqrt{N}}+\frac{{\int }_0^{t}g(s)ds}{N},$$

$$A(\varphi )=a(\overline{\varphi })-a^{\ast }(\varphi )$$

$$H_N^{\prime }$$

$$\frac{1}{i}\frac{\partial }{\partial t}\Psi =(\tilde{L}-N{\chi }_0-{\chi }_1)\Psi ,$$

$$\Vert \tilde{L}\Omega {\Vert }_{\mathcal{F}}\le N^{-1/2}\Vert e^B[A,V]e^{-B}\Omega {\Vert }_{\mathcal{F}}+N^{-1}\Vert e^{B}V{e}^{-B}\Omega {\Vert }_{\mathcal{F}}\text{ ,}$$

$$\Psi (t)=e^{B(t)}{e}^{\sqrt{N}A(t)}{e}^{itH}{e}^{-\sqrt{N}A(0)}\Omega .$$

$$\left(\frac{1}{i}\frac{\partial }{\partial t}-\tilde{L}\right)(e^{i{\int }_0^t(N{\chi }_0(s)+{\chi }_1(s))ds}\Psi -\Omega )=\tilde{L}\Omega ,$$

$$H_N$$

$$=:H_0+\frac{1}{6N^2}V,$$

$$H_{N,N}=\sum _{j=1}^n{\Delta }_{x_j}+\frac{1}{N^2}\sum _{i<j<k}^{n}v(x_i-x_j,x_j-x_k).$$

$$i\frac{\partial }{\partial t}\varphi +△\varphi +\frac{1}{2}\varphi \int v(x-y,y-z){\left|\varphi (y)\right|}^2{\left|\varphi (z)\right|}^{2}dydz=0$$

$$\displaystyle g(t,x,y):=-\triangle\delta_{12}-\bigg{(}\int v_{12,23}\left|\phi_{3}\right|^{2}dz\bigg{)}\overline{\phi}_{1}\phi_{2}-\frac{1}{2}\bigg{(}\int v_{12,23}\left|\phi_{2}\right|^{2}\left|\phi_{3}\right|^{2}dydz\bigg{)}\delta_{12}$$

$$\displaystyle m(t,x,y):=\bigg{(}\int v_{12,23}\left|\phi_{3}\right|^{2}dz\bigg{)}\overline{\phi}_{1}\overline{\phi}_{2}$$

$$h(t):=\Vert e^B[A,[A,V]]e^{-B}\Omega {\Vert }_{\mathcal{F}}$$

$$i(t):=\Vert e^B[A,[A,[A,V]]]e^{-B}\Omega {\Vert }_{\mathcal{F}}$$

$$\Vert e^{-\sqrt{N}A(t)}{e}^{-B(t)}{e}^{-i{\int }_0^t(N{\chi }_0(s)+{\chi }_1(s))ds}\Omega -e^{it{H}_N}{e}^{-\sqrt{N}A(0)}\Omega {\Vert }_{\mathcal{F}}$$

$$\le \frac{{\int }_0^{t}f(s)ds}{6N^{\frac{3}{2}}}+\frac{{\int }_0^{t}g(s)ds}{6N^2}+\frac{{\int }_0^{t}h(s)ds}{12N}+\frac{{\int }_0^{t}i(s)ds}{36N^{\frac{1}{2}}}.$$

$$(i{u}_t+u{g}^T+gu-(I+p)m)=(i{p}_t+[g,p]+u\overline{m})(1+p{)}^{-1}u.$$

$$\tilde{L}=$$

$$\frac{1}{2}\int v_{12,23}{\left|{\varphi }_2\right|}^2{\left|{\varphi }_3\right|}^2{a}_x^{\ast }{a}_{x}dxdydz$$

$$-$$

$$\frac{1}{6N^{3/2}}{e}^B[A,V]e^{-B}+\frac{1}{12N}{e}^B[A,[A,V]]e^{-B}$$

$$+\frac{1}{36N^{\frac{1}{2}}}{e}^B[A,[A,[A,V]]]e^{-B},$$

$$[A,V]$$

$$[A,[A,V]]$$

$$[A,[A,[A,V]]]$$

Full text

The author thanks Matei Machedon for discussions related to this work.

1. Introduction

Recently, a second-order correction to the usual tensor product (mean-field) approximation for the Hamiltonian evolution of a many-particle system with 2-body potential was found in Grillakis-Machedon-Margetis[1]. Here is their main result,

Theorem 1**.**

[1]** Let $\phi$ be a smooth solution of the Hartree equation

[TABLE]

with initial conditions $\phi_{0}$ and potential $v^{\prime}$ being even. Assuming the following:

1. (1)

Let $k(t,x,y)\in L_{s}^{2}(dxdy)$ for a.e. $t$, solve the Grillakis-Machedon-Margetis equation

[TABLE]

where products in equation 1.2 mean compositions of operators, and

[TABLE] 2. (2)

Functions $f(t):=\|e^{B}[A,V^{\prime}]e^{-B}\Omega\|_{{\mathcal{F}}}$ and $g(t):=\|e^{B}V^{\prime}e^{-B}\Omega\|_{{\mathcal{F}}}$ are locally integrable, where

[TABLE]

and ${\mathcal{F}}$ the Fock space defined in **[1]**. 3. (3)

$\int d(t,x,x)\ dx$* is locally integrable in time, where*

[TABLE]

Then, there exist real functions $\chi_{0}$, $\chi_{1}$ such that

[TABLE]

where $\Omega=(1,0,0,\cdots)\in{\mathcal{F}}$ is the vacuum state,

[TABLE]

with $a$ and $a^{\ast}$ are the annihilation operator and creation operator, and the Hamiltonian

[TABLE]

To prove Theorem 1, the key is to notice that, for $\phi$ and $k$ satisfying the assumptions, there is some ”nice” Hermitian $\widetilde{L}$ such that

[TABLE]

and

[TABLE]

where

[TABLE]

Then applying the energy estimate to

[TABLE]

would give the estimate 1.5.

With the same set up, this result can be extended to the case of 3-body interaction potentials.

2. Main Result

We consider the Fock space Hamiltonian $H_{N}:{\mathcal{F}}\rightarrow{\mathcal{F}}$ with the 3-body interaction potential defined by

[TABLE]

in correspondence to the Hamiltonian

[TABLE]

We assume that $v$ is symmetric in $x$, $y$, and $z$. As shown in (1.2) and (1.3) in Chen-Pavlović[2], every translation invariant 3-body potential can be written in the form $v(x-y,y-z)$. However, I don’t assume $v\geqslant 0$ as in [2].

Theorem 2**.**

With the same assumptions as in theorem 1 with $H_{N}^{\prime}$ replaced by $H_{N}$, $V^{\prime}$ by $V$ in formula 2.1. We change equation 1.1 to

[TABLE]

and

[TABLE]

where $v_{12,23}\left|\phi_{3}\right|^{2}$ is an abbreviation for the product $v(x-y,y-z)\left|\phi(z)\right|^{2}$,

Further, we assume that the functions

[TABLE]

and

[TABLE]

are also locally integrable.

Then, there exist real functions $\chi_{0}$, $\chi_{1}$ such that

[TABLE]

Proof.

Similar to the proof of Theorem 1.

Remark 1**.**

Recall that, the equation for $k$ is 1.2

[TABLE]

A derivation of this equation will be given in Section 4.

Remark 2**.**

In Section 4, we will show that the $\widetilde{L}$ in this case is

[TABLE]

where $d(t,x,y)$ is the same as equation 1.4.

3. The New Hartree equation

In this section we derive the Hartree equation 2.2 for the one-particle wave function $\phi$ as needed in Theorem 2.

Lemma 1**.**

The following relations hold, where $A$ denotes $A(\phi)$, and $A$ , $V$ are defined by formulas 1.6 and 2.1):

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Proof.

This is a direct calculation.

Now, we write $\Psi_{1}(t)=e^{\sqrt{N}A(t)}e^{itH}e^{-\sqrt{N}A(0)}\Omega$ for which we carry out the calculation in the spirit of equation (3.7) in Rodnianski-Schlein [3].

Proposition 1**.**

Say $\phi$ satisfies the Hartree equation

[TABLE]

then $\Psi_{1}(t)$ satisfies

[TABLE]

Proof.

Applying the formulas

[TABLE]

[TABLE]

to $C=\sqrt{N}A$ we have

[TABLE]

where

[TABLE]

The Hartree equation2.2 is equivalent to setting

[TABLE]

Or more explicitly, equation 3.2 is

[TABLE]

because of Lemma 1 and the fact that $[\Delta_{x}a_{x},a_{y}^{\ast}]=(\Delta\delta)(x-y)$.

Thus

[TABLE]

i.e 3.1 simplifies to

[TABLE]

Write the last term as

[TABLE]

then the first two terms on the right-hand side of 3.3 are the main ones we need to consider, since the next four terms are $O\left(\frac{1}{N^{2}}\right)$, $O\left(\frac{1}{N^{\frac{3}{2}}}\right),$ $O\left(\frac{1}{N}\right),$and $O\left(\frac{1}{N^{\frac{1}{2}}}\right)$.

In order to kill the terms involving ”only creation operators” i.e $a_{x}^{\ast}a_{y}^{\ast}$ in $\frac{1}{4!}\frac{1}{6}[A,[A,[A,[A,V]]]]$, we introduce $B$ (see 1.3) and let

[TABLE]

as in [1]. So we have

[TABLE]

where

[TABLE]

and

[TABLE]

4. Equation for $k$

In this section, we derive the equation for $k$.

Let ${\mathcal{I}}:sp\rightarrow Quad$ be the Lie algebra isomorphism defined by $Q(d,k,l)={\mathcal{I}}(S(d,k,l))$ in theorem 3 in [1], where $sp$ is the infinite dimensional Lie algebra containing

[TABLE]

and $k$ and $l$ are symmetric, and $Quad$ is the Lie algebra consisting of quadratics of the form

[TABLE]

Recall that

[TABLE]

we write

[TABLE]

with

[TABLE]

and

[TABLE]

Though it’s true that

[TABLE]

for

[TABLE]

it’s not quite true that

[TABLE]

However, the commutators of ${\mathcal{I}}(G+M)$ and $H_{0}+\frac{1}{4!}\frac{1}{6}[A,[A,[A,[A,V]]]]$ with $B$ are the same as in the discussion in page 15 in [1].

Now, $L_{Q}$ reads

[TABLE]

where

[TABLE]

in this setting.

Then by the definition of the isomorphism ${\mathcal{I}}$, the coefficient of $a_{x}a_{y}$ is $-(\mathcal{M}_{1,12}+\mathcal{M}_{2,12}+\mathcal{M}_{3,12}),$ and the coefficient of $a_{x}^{\ast}a_{y}^{\ast}$ is $(\mathcal{M}_{1,21}+\mathcal{M}_{2,21}+\mathcal{M}_{3,21}).$ To write it explicitly:

[TABLE]

Setting formula 4.2 to [math] gives the equation 2.3. Hence we have the following theorem similar to Corollary 1 in [1].

Theorem 3**.**

If $\phi$ and $k$ solve 2.2 and 2.3, then the coefficients of $a_{x}a_{y}$ and $a_{x}^{\ast}a_{y}^{\ast}$ drop out and $L_{Q}$ becomes

[TABLE]

where $d$ is given by formula 1.4 and the full operator reads

[TABLE]

and

[TABLE]

[TABLE]

This matches equation 2.4.

Notice that

[TABLE]

in our case. This ends the proof of theorem 2.

The verification that the hypotheses of the main theorem are satisfied by some singular potentials will be provided in a subsequent article.