Lieb–Robinson Bounds for the Toda Lattice

Abstract

We establish locality estimates, known as Lieb–Robinson bounds, for the Toda lattice. In contrast to harmonic models, the Lieb–Robinson velocity for these systems do depend on the initial condition. Our results also apply to the entire Toda as well as the Kac-van Moerbeke hierarchy. Under suitable assumptions, our methods also yield a finite velocity for certain perturbations of these systems.

Appendix Existence of Bounded Solutions for Hamiltonian Systems

In this appendix we want to look at a general Hamiltonian system with nearest neighbor interaction:

$$ H(\mathrm{x})=\sum_{n\in \mathbb {Z}} \biggl( \frac{p_n^2}{2} + V(q_{n+1} -q_{n}) \biggr), $$

(A.1)

where x={(p _n ,q _n )} _n∈ℤ. Let V∈C ²(ℝ) with V(x)≥0 and V(0)=V′(0)=0 such that 0 is a fixed point of the system. Since we want to obtain bounded solutions we will assume that our interaction potential is confining in the sense that V(x)→+∞ as x→±∞. Since the uniform motion q _n (t)=q ⁰+p ⁰ t (with q ⁰,p ⁰ some real constants) of the system is unbounded we switch to relative coordinates r _n =q _n+1−q _n in which the equation of motions read

$$ \dot{r}_n = p_{n+1} - p_n, \qquad\dot{p}_n = V'(r_n) - V'(r_{n-1}). $$

(A.2)

We will consider this system in the Hilbert space $\mathcal{X}=\ell^{2}(\mathbb {Z}) \times\ell^{2}(\mathbb {Z})$.

Theorem A.1

Suppose V∈C ²(ℝ) such that 0 is a unique global minimum with V(0)=V′(0)=0, V″(0)>0 and V(x)→+∞ as |x|→∞.

Then the system (A.2) has a unique global solution in $\mathcal{X}$ for which the energy

$$ H(p,r)= \sum_{n\in \mathbb {Z}} \biggl( \frac{p_n^2}{2} + V(r_n) \biggr) $$

(A.3)

is finite and conserved. This solution is C ¹ with respect to the initial condition. Moreover, 0 is a stable fixed point and all solutions satisfy ∥(p(t),r(t))∥₂≤C as well as ∥(p(t),r(t))∥_∞≤C, where the constant C depends only on the initial condition.

Proof

First of all note that by our assumption on V we can find constants c _R and C _R for every R>0 such that |V(x)|≤C _R x ², |V′(x)|≤C _R |x| and |V(x)|≥c _R x ² for |x|≤R.

In particular, for r∈ℓ ²(ℤ) with ∥r∥₂≤R we have ∥V′(r)∥₂≤C _R ∥r∥₂ and it follows that the map r _n ↦V′(r _n ) is C ¹ on ℓ ²(ℤ). Since the same is true for the shift operator x _n ↦x _n−1, our vector field is C ¹ and local existence and uniqueness follow from standard results [1, Thm. 4.1.5]. This also implies that the flow is C ¹ with respect to the initial condition [1, Lem. 4.1.9]. Moreover, |H(p,r)|≤C _R ∥(p,r)∥₂ implies that H is finite on $\mathcal{X}$ and a straightforward calculation shows that it is conserved by the flow.

Moreover, H(p,r)≥c ₁  ε for ∥(p,r)∥₂=ε and ε≤1. Hence [1, Thm. 4.3.11] shows that 0 is a stable fixed point.

Finally, if V(x)→+∞ there is a constant M _E such that |x|≤M _E whenever |V(x)|≤E. Hence setting E=H(p(0),r(0)) we have H(p(t),r(t))=E implying $\|p(t)\|_{2} \le\sqrt{2E}$ and ∥r(t)∥_∞≤M _E . But this implies $\|r(t)\|_{2} \le c_{M_{E}}^{-1} \|V(r)\|_{2} \le c_{M_{E}}^{-1} \sqrt{E}$. Hence our vector field remains bounded along integral curves on finite t intervals and hence all solutions are global in time by [1, Prop. 4.1.22]. □

Note that, using $q_{n}(t) =q_{n}(0) + \int_{0}^{t} p_{n}(s) ds$, for our original variables we get

$$ \bigl\|q(t)\bigr\|_2 \le \bigl\|q(0)\bigr\|_2 + C |t|, \qquad \bigl\|q(t) \bigr\|_\infty\le \bigl\|q(0)\bigr\|_\infty+ C |t|. $$

(A.4)

Moreover, clearly the Toda potential V(r)=e ^−r+r−1 satisfies the above assumptions.

Theorem A.2

Let $\mathrm {x}=(p,r) \in\mathcal{X}$ and μ>0. There exists a number v=v(μ,x) for which given any n,m∈ℤ, the bound

$$ \max \biggl[ \biggl \vert \frac{\partial }{ \partial z} p_n(t)\biggr \vert , \biggl \vert \frac{\partial }{ \partial z} r_n(t)\biggr \vert \biggr] \leq C \, e^{- \mu( |n-m| - v |t| )} , $$

(A.5)

holds for all t∈ℝ and each z∈{p _m ,r _m }, where

$$ C = C(\mathrm {x}) = \max \Bigl( \sup_{(t,n)\in \mathbb {R}\times \mathbb {Z}} \bigl|V' \bigl(r_n(t) \bigr)\bigr|^{1/2}, 1 \Bigr). $$

(A.6)

In fact, one may take

$$ v = 2 C \biggl(e^{\mu+1}+\frac{1}{\mu} \biggr). $$

(A.7)

Proof

Fix $\mathrm {x}\in\mathcal{X}$. Our previous theorem guarantees that for each x∈M and n∈ℤ, the function F _n :ℝ→ℝ² given by

$$ F_n(t; \mathrm {x})= \begin{pmatrix} r_n(t) \\[3pt] p_n(t) \end{pmatrix} $$

(A.8)

is well-defined and differentiable with respect to each z∈{p _m ,r _m }. When convenient, we will suppress the dependence of F _n on x. Using the equations of motion, i.e. (A.2), it is clear that

$$ F_n(t)= F_n(0)+\int_0^t \begin{pmatrix} p_{n+1}(s) - p_n(s) \\[3pt] V'(r_n(s)) - V'(r_{n-1}(s)) \end{pmatrix} \, ds . $$

(A.9)

Differentiating with respect to z we obtain

$$ \frac{\partial }{ \partial z} F_n(t) = \frac{\partial }{ \partial z} F_n(0) + \sum_{|e|\leq 1}\int _0^t D_{n,e}(s) \frac{\partial }{ \partial z} F_{n+e}(s) \, ds, $$

(A.10)

with

$$ D_{n,e}(s) = \begin{pmatrix}0 & \delta_1(e)-\delta_0(e) \\[3pt] V'(r_n) \delta_0(e) - V'(r_{n-1}) \delta_{-1}(e) & 0 \end{pmatrix} . $$

(A.11)

Hence

$$ \biggl\vert\frac{\partial }{ \partial z} F_n(t)\biggr\vert\leq \biggl\vert\frac{\partial }{ \partial z} F_n(0)\biggr\vert+ \sum_{|e|\leq1}\int _0^{|t|} D_{e} \biggl\vert\frac{\partial }{ \partial z} F_{n+e}(s)\biggr\vert\, ds, $$

(A.12)

where

$$ D_e = \begin{pmatrix} 0 & \quad \delta_1(e)+\delta_0(e) \\[3pt] C^2 (\delta_0(e) + \delta_{-1}(e)) &\quad 0 \end{pmatrix} . $$

(A.13)

Let us now consider the case that z=r _m , i.e.,

$$ \frac{ \partial}{\partial r_m}F_n(0) = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \delta_m(n) . $$

(A.14)

In this case, iteration yields

(A.15)

where we have set

$$ D = \frac{1}{2C} \sum_{|e| \leq1} D_e = \begin{pmatrix} 0 & \quad C^{-1} \\[3pt] C &\quad 0 \end{pmatrix} . $$

(A.16)

Again, convergence is guaranteed since $\frac{\partial }{ \partial z} F_{n}(t)$ is continuous and thus bounded on compact time intervals. Using , one obtains

$$ \biggl \Vert \frac{\partial }{\partial r_m}F_n(t) \biggr \Vert _{\infty} \leq C \sum_{k=|n-m|}^\infty \frac{|2C t|^k}{k!} $$

(A.17)

The rest follows as in Theorem 2.1. □

Note that in Flaschka variables (2.3) the equations of motion read

$$ \begin{array}{c} \dot{a}_n(t) = a_n(t) \bigl( b_{n+1}(t) - b_n(t) \bigr) \quad\mbox{and} \\[5pt] \dot{b}_n(t) = -\dfrac{1}{2} \bigl(V' \bigl(- \ln \bigl(4a_n^2 \bigr) \bigr) - V' \bigl(- \ln \bigl(4a_{n-1}^2 \bigr) \bigr) \bigr) , \end{array} $$

(A.18)

and (p,r) will be bounded if and only if (a,a ⁻¹,b) are bounded. The fixed point in these new coordinates is $(a_{0},b_{0})=(\frac{1}{2},0)$ and $(p,r)\in\mathcal{X}$ if and only if $(a,b)-(a_{0},b_{0})=\allowbreak (a-\frac{1}{2},b) \in\mathcal{X}$.

Finally, by

$$ a_n(t) = a_n(0) \exp \biggl(\,\int _0^t \bigl(b_{n+1}(s) - b_n(s) \bigr)\, ds \biggr) $$

(A.19)

the sign of a _n is preserved under the flow.