Correction to: Lie–Poisson Methods for Isospectral Flows

Correction to: Foundations of Computational Mathematics (2020) 20:889–921 https://doi.org/10.1007/s10208-019-09428-w

Motivation The authors of [1] have found an error in the proof of Proposition 1. Specifically, the proof presented there implicitly assumed that the matrices $\nabla H(Q(t)^\dagger P(t))^\dagger$ commute for any $t\ge 0$. The statement of the Proposition 1 has now been reformulated in a suitable way to fit with the discussion below Proposition 1 in [1]. Here below, we present the corrected version and proof of Proposition 1.

FormalPara Proposition 1

Consider a solution (Q(t), P(t)) of Hamilton’s equations (12) defined for $0\le t\le T$ and a given initial point $(Q_0, P_0)$, such that $Q_0^\dagger P_0\in S$, for $S\subseteq \mathfrak {gl}(n,\mathbb {C})^*$ a linear subspace as before, and $Q_0$ is invertible. Then, there exists $G\in GL(n,\mathbb {C})$ such that $Q(t)^\dagger G\in N(S)$, for any $0\le t\le T$,¹ if and only if $\nabla H(Q(t)^\dagger P(t))^\dagger \in \mathfrak {n}(S)$, for any $0\le t\le T$. Furthermore, if one of the two condition holds, then $Q^\dagger (t)P(t)\in S$, for any $0\le t\le T$.

FormalPara Proof

Let us first prove the equivalence of the two conditions. Let us assume that $\nabla H(Q(t)^\dagger P(t))^\dagger \in \mathfrak {n}(S)$, for any $0\le t\le T$. We have

$$\begin{aligned} Q(t)^\dagger G=\exp \left( \int _{0}^{t} \Theta (s) ds\right) Q_0^\dagger G, \end{aligned}$$

where $\Theta$ is a solution to

$$ \frac{d\Theta (t)}{dt}=\text{ dexp}^{-1}_{\Theta (t)}\nabla H(Q(t)^\dagger P(t))^\dagger \quad \Theta (0)=0, $$

where $\text{ dexp}^{-1}$ is defined as in [2]. Hence, $\Theta (t)\in \mathfrak {n}(S)$, for any $0\le t\le T$, being defined as the Magnus expansion of $\nabla H(Q(t)^\dagger P(t))^\dagger$ (see [2]). This proves the statement, since $N(S)\supseteq \exp (\mathfrak {n}(S))$.

Conversely, let us assume that there exists $G\in GL(n,\mathbb {C})$ such that $Q(t)^\dagger G\in N(S)$ for any $0\le t\le T$. By the formula above, we have

$$\begin{aligned} Q(t)^\dagger Q_0^{-\dagger }=\exp \left( \int _{0}^{t}\Theta (s) ds\right) . \end{aligned}$$

Since the left-hand side is in N(S) for any $0\le t\le T$, we have $\Theta (t)\in \mathfrak {n}(S)$. This implies that also $\frac{d\Theta (t)}{dt}\in \mathfrak {n}(S)$. Therefore, since $\text{ dexp}_{\Theta (t)}\frac{d\Theta (t)}{dt}=\nabla H(Q^\dagger P)^\dagger$, we get the thesis.

Finally, we have

$$\begin{aligned} Q(t)^{\dagger } P(t)=\exp \left( \int _{0}^{t}\Theta (s) ds\right) Q_0^\dagger P_0\exp \left( -\int _{0}^{t}\Theta (s) ds\right) . \end{aligned}$$

Hence, if one of the two condition holds, $Q^\dagger (t)P(t)\in S$, for any $0\le t\le T$, by the definition of N(S). $\square$