Sobolev-type spaces

For the study of the generalized half-wave maps equations (HWM $_d$ ), we introduce the following Sobolev-type spaces for matrix-valued functions. For an integer $d \geq 2$ , we use $M_d(\mathbb {C}) \equiv \mathbb {C}^{d \times d}$ to denote the Hilbert space of complex $d \times d$ -matrices equipped with the inner product

$$ \begin{align*}\langle A, B \rangle_F := \mathrm{Tr}( A B^*) \quad \text{for } A,B \in M_d(\mathbb{C}) \end{align*} $$

and the corresponding Frobenius norm of a matrix $A \in M_d(\mathbb {C})$ will be denoted by $| A |_F = \sqrt {\langle A, A \rangle _F}$ .

The Lebesgue spaces $L^p(\mathbb {R}; M_d(\mathbb {C}))$ and $L^p_{\mathrm {loc}}(\mathbb {R}; M_d(\mathbb {C}))$ are defined in an obvious manner. For $s> 0$ , we use the Sobolev spaces

$$ \begin{align*}\dot{H}^s(\mathbb{R}; M_d(\mathbb{C})) := \left \{ \mathbf{U} \in L^1_{\mathrm{loc}}(\mathbb{R}; M_d(\mathbb{C})) \mid \| \mathbf{U} \|_{\dot{H}^s} := \| |D|^s \mathbf{U} \|_{L^2} < +\infty \right \} \, , \end{align*} $$

$$ \begin{align*}H^s(\mathbb{R}; M_d(\mathbb{C})) := \left \{ \mathbf{V} \in L^1_{\mathrm{loc}}(\mathbb{R}; M_d(\mathbb{C})) \mid \| \mathbf{V} \|_{H^s} := \| \langle D \rangle^s \mathbf{V} \|_{L^2} < +\infty \right \} \,. \end{align*} $$

We set $\dot {H}^\infty (\mathbb {R}; M_d(\mathbb {C})) := \cap _{s> 0} \dot {H}^s(\mathbb {R}; M_d(\mathbb {C}))$ and $H^\infty (\mathbb {R}; M_d(\mathbb {C})) := \cap _{s>0} H^s(\mathbb {R}; M_d(\mathbb {C}))$ . Note that $\| \cdot \|_{\dot {H}^s}$ is a semi-norm, since nontrivial constant maps also belong to $\dot {H}^s(\mathbb {R}; M_d(\mathbb {C}))$ for $s>0$ . Furthermore, for $0 \leq k \leq d$ given, we define the spaces

$$ \begin{align*}\dot{H}^s(\mathbb{R}; \mathsf{Gr}_k(\mathbb{C}^d)) := \left \{ \mathbf{U} \in \dot{H}^s(\mathbb{R}; M_d(\mathbb{C})) \mid \mathbf{U}(x) \in \mathsf{Gr}_k(\mathbb{C}^d) \text{ for a. e. } x \in \mathbb{R} \right \} \,. \end{align*} $$

Note that the scaling-critical energy space associated to (HWM $_d$ ) with target $\mathsf {Gr}_k(\mathbb {C}^d)$ is $\dot {H}^{\frac 1 2}(\mathbb {R}; \mathsf {Gr}_k(\mathbb {C}^d))$ equipped with the Gagliardo semi-norm $\| \cdot \|_{\dot {H}^{\frac 1 2}}$ such that

(2.1) $$ \begin{align} \| \mathbf{U} \|_{\dot{H}^{\frac 1 2}}^2 = \| |D|^{\frac 1 2} \mathbf{U} \|_{L^2}^2 = \frac{1}{2 \pi} \int_{\mathbb{R}} \! \int_{\mathbb{R}} \frac{|\mathbf{U}(x)- \mathbf{U}(y)|_F^2}{|x-y|^2} \, dx \, dy \,. \end{align} $$

Note that $E(\mathbf {U}) = \frac {1}{2} \| \mathbf {U} \|_{\dot {H}^{\frac 1 2}}^2$ is the Hamiltonian energy functional for (HWM $_d$ ) with the natural symplectic form for maps defined on $\mathbb {R}$ with values in the complex Grassmannian $\mathsf {Gr}_k(\mathbb {C}^d)$ .

In addition to the space $\dot {H}^s$ -spaces, it turns out to be convenient to introduce the following family of affine inhomogeneous Sobolev-type spaces given by

$$ \begin{align*}H^s_\bullet(\mathbb{R}; M_d(\mathbb{C})) := M_d(\mathbb{C}) \oplus H^s(\mathbb{R}; M_d(\mathbb{C})) \end{align*} $$

and we define $H^\infty _\bullet (\mathbb {R}; M_d(\mathbb {C})) := \cap _{s> 0} H^s_\bullet (\mathbb {R};M_d(\mathbb {C}))$ . Furthermore, we set

$$ \begin{align*}H^s_\bullet(\mathbb{R}; \mathsf{Gr}_k(\mathbb{C}^d)) := \left \{ \mathbf{U}\in H^s_\bullet(\mathbb{R}; M_d(\mathbb{C})) \mid \mathbf{U}(x) \in \mathsf{Gr}_k(\mathbb{C}^d) \text{ for a. e. } x\in \mathbb{R} \right \} \,. \end{align*} $$

It is easy to see that the following strict inclusions hold true:

$$ \begin{align*}\mathcal{R}at(\mathbb{R}; \mathsf{Gr}_k(\mathbb{C}^d)) \subsetneq H^s_\bullet(\mathbb{R}; \mathsf{Gr}_k(\mathbb{C}^d)) \subsetneq \dot{H}^s(\mathbb{R}; \mathsf{Gr}_k(\mathbb{C}^d)) \subsetneq L^\infty(\mathbb{R}; M_d(\mathbb{C})) \, , \end{align*} $$

where we recall that $\mathcal {R}at(\mathbb {R}; \mathsf {Gr}_k(\mathbb {C}^d))$ denotes the space of rational maps from $\mathbb {R}$ to $\mathsf {Gr}_k(\mathbb {C}^d)$ .

For (HWM) with maps valued in the unit sphere $\mathbb {S}^2 \subset \mathbb {R}^3$ , we make use of the corresponding Sobolev spaces $H^s(\mathbb {R}; \mathbb {R}^3)$ and $\dot {H}^s(\mathbb {R}; \mathbb {R}^3)$ , where the energy space is

$$ \begin{align*}\dot{H}^{\frac 1 2}(\mathbb{R}; \mathbb{S}^2) := \left \{ \mathbf{u} \in \dot{H}^{\frac 1 2}(\mathbb{R}; \mathbb{R}^3) \mid \mathbf{u}(x) \in \mathbb{S}^2 \text{ for a. e. } x \in \mathbb{R} \right \} \end{align*} $$

endowed with the Gagliardo semi-norm $\| \cdot \|_{\dot {H}^{\frac 1 2}}$ such that

$$ \begin{align*}\| \mathbf{u} \|_{\dot{H}^{\frac 1 2}}^2 = \| |D|^{\frac 1 2} \mathbf{u} \|_{L^2}^2 = \frac{1}{2 \pi} \int_{\mathbb{R}} \! \int_{\mathbb{R}} \frac{|\mathbf{u}(x)- \mathbf{u}(y)|^2}{|x-y|^2} \, dx \, dy \,. \end{align*} $$

From the introduction above, we recall that unit vectors $\mathbf {u} \in \mathbb {S}^2$ can be equivalently encoded by using the standard Pauli matrices $(\sigma _1, \sigma _2, \sigma _3)$ via the relation

$$ \begin{align*}\mathbf{U} = \mathbf{u} \cdot \boldsymbol{\sigma} = u_1 \sigma_1 +u_2 \sigma_2 + u_3 \sigma_3 = \left ( \begin{array}{cc} u_3 & u_1 - \mathrm{i} u_2 \\ u_1 + \mathrm{i} u_2 & -u_3 \end{array} \right ) \, , \end{align*} $$

where we easily check that $\mathbf {U} = \mathbf {U}^*$ with and $\mathrm {Tr}(\mathbf {U}) = 0$ . Also, we find that $u_k= \frac {1}{2} \mathrm {Tr}(\mathbf {U} \sigma _k) = \frac {1}{2} \langle \mathbf {U}, \sigma _k \rangle _F$ for $k=1,2,3$ . Thus, by means of the relation $\mathbf {U} = \mathbf {u} \cdot \boldsymbol {\sigma }$ , we find the equivalence of (semi)-norms $\| \mathbf {u} \|_{\dot {H}^{s}} \sim \| \mathbf {U} \|_{\dot {H}^{s}}$ for all $s>0$ .

Hardy spaces, Toeplitz and Hankel operators

We consider the Hilbert space $L^2(\mathbb {R}; \mathcal {V})$ for maps on $\mathbb {R}$ with values in the finite-dimensional Hilbert spaces

$$ \begin{align*}\mathcal{V} = \mathbb{C}^d \quad \text{or} \quad \mathcal{V} = M_d(\mathbb{C}) \, , \end{align*} $$

which we endow with their natural inner products and norms, that is,

$$ \begin{align*}\langle u,v \rangle_{\mathcal{V}} = \sum_{k=1}^d u_k \overline{v}_k \quad \text{if } \mathcal{V}= \mathbb{C}^d, \qquad \langle A,B \rangle_{\mathcal{V}} = \mathrm{Tr}(A B^*) \quad \text{if } \mathcal{V} = M_d(\mathbb{C}) \,. \end{align*} $$

The Cauchy–Szegő projection $\Pi _+ : L^2(\mathbb {R}; \mathcal {V}) \to L^2_+(\mathbb {R};\mathcal {V})$ onto the Hardy space

$$ \begin{align*}L^2_+(\mathbb{R}; \mathcal{V}) := \{ f \in L^2(\mathbb{R}; \mathcal{V}) \mid \mathrm{supp} \, \widehat{f} \subset [0, \infty) \} \end{align*} $$

is given by

$$ \begin{align*}(\Pi_+ f)(x) := \frac{1}{2 \pi} \int_0^{+\infty} \mathrm{e}^{\mathrm{i} x \xi} \widehat{f}(\xi) \, d\xi \quad \text{with} \quad \widehat{f}(\xi) = \int_{-\infty}^{+\infty} f(x) \mathrm{e}^{-\mathrm{i} \xi x} \, dx \, , \end{align*} $$

or, equivalently, we have on the Fourier side. We use

$$ \begin{align*}\Pi_- := \mathrm{Id} - \Pi_+ \end{align*} $$

to denote projection onto the orthogonal complement

$$ \begin{align*}L^2_-(\mathbb{R}; \mathcal{V}) := (L^2_+(\mathbb{R};\mathcal{V}))^\perp = \{ f \in L^2(\mathbb{R};\mathcal{V}) \mid \mathrm{supp} \, \widehat{f}(\xi) \subset (-\infty,0] \}\,. \end{align*} $$

From standard Paley–Wiener theory we recall that elements $f \in L^2_+(\mathbb {R}; \mathcal {V})$ can be naturally identified with holomorphic functions defined on the complex upper half-plane $\mathbb {C}_+$ such that

$$ \begin{align*}L^2_+(\mathbb{R}; \mathcal{V}) \cong\left \{ f \in \mathrm{Hol}(\mathbb{C}_+; \mathcal{V}) \mid \sup_{y>0} \int_{\mathbb{R}} |f(x+ \mathrm{i} y)|_{\mathcal{V}} ^2 \, dx < +\infty \, \right \} \, , \end{align*} $$

where $|\cdot |_{\mathcal {V}}$ denotes the norm on $\mathcal {V}$ . Throughout this paper, we will freely make use of this fact and we thus regard elements $f \in L^2_+(\mathbb {R}; \mathcal {V})$ as holomorphic functions $f=f(z)$ with $z \in \mathbb {C}_+$ . We use $\mathsf {H} = -\mathrm {i} \Pi _+ + \mathrm {i} \Pi _-$ to denote the Hilbert transform on $L^2(\mathbb {R};\mathcal {V})$ , which can also be written as the singular integral operator

$$ \begin{align*}(\mathsf{H} f)(x) = \frac{1}{\pi} \mathrm{p.v.} \int_{\mathbb{R}} \frac{f(y)}{x-y} \, dy. \end{align*} $$

For a bounded matrix-valued function $\mathbf {U} \in L^\infty (\mathbb {R}; M_d(\mathbb {C}))$ , we define the corresponding Toeplitz operator as

$$ \begin{align*}T_{\mathbf{U}} : L^2_+(\mathbb{R}; \mathcal{V}) \to L^2_+(\mathbb{R};\mathcal{V}), \quad f \mapsto T_{\mathbf{U}} f := \Pi_+(\mathbf{U} f) \,. \end{align*} $$

Likewise, the corresponding Hankel operator is given by

$$ \begin{align*}H_{\mathbf{U}} : L^2_+(\mathbb{R}; \mathcal{V}) \to L^2_-(\mathbb{R}; \mathcal{V}), \quad f \mapsto H_{\mathbf{U}} f := \Pi_-(\mathbf{U} f ) \,. \end{align*} $$

We remark that we adapt the definition of $H_{\mathbf {U}}$ from Peller’s book [Reference Peller32]; another (equivalent) definition of Hankel operators can be achieved by anti-linear operators (see, e.g., [Reference Gérard and Pushnitski17]). However, for studying the Lax pair structure for (HWM $_d$ ), we have found it more convenient to use the present convention for $H_{\mathbf {U}}$ .

A central fact about Hankel operators used in this paper is Kronecker’s theorem, which relates the rationality of the symbol $\mathbf {U}$ with the property that $H_{\mathbf {U}}$ has finite rank. We refer the reader to Section 4 for details. Furthermore, we remark that $H_{\mathbf {U}}$ is Hilbert-Schmidt if and only if $\mathbf {U} \in \dot {H}^{\frac 1 2}$ ; see again Section 4 for a detailed discussion.

The operators $X^*$ , X, and $I_+$

On the Hardy space $L^2_+(\mathbb {R}; \mathcal {V})$ , we recall that the adjoint Lax–Beurling semigroup $\{ S(\eta )^* \}_{\eta \geq 0}$ is given by

$$ \begin{align*}(S(\eta)^* f)(x) = \Pi_+( e^{-\mathrm{i} \eta x} f(x)) \quad \text{for } f \in L^2_+(\mathbb{R}; \mathcal{V}) \text{ and } \eta \geq 0 \, , \end{align*} $$

which corresponds to the contraction semigroup of left shifts on $L^2_+(\mathbb {R}; \mathcal {V})$ . We remark that $S(\eta )^* = e^{-\mathrm {i} \eta X^*}$ , where its generator $X^*$ is given by the unbounded operator

with the operator domain

$$ \begin{align*}\mathrm{dom}(X^*) = \big \{ f \in L^2_+(\mathbb{R}; \mathcal{V}) \mid \frac{d \widehat{f}}{d\xi} \in L^2(\mathbb{R}_+; \mathcal{V}) \big \} \,. \end{align*} $$

It is straightforward to check that all rational functions $f\in L^2_+(\mathbb {R}; \mathcal {V})$ belong to $\mathrm {dom}(X^*)$ . For $z_0 \in \mathbb {C}_+$ , the action of the resolvent $(X^*-z_0)^{-1}$ is easily found to be

$$ \begin{align*}((X^*-z_0)^{-1} f)(z) = \frac{f(z)-f(z_0)}{z-z_0} \, , \end{align*} $$

for all $f \in L^2_+(\mathbb {R}; \mathcal {V})$ . We remark that $X^*$ is the adjoint of the unbounded operator

$$ \begin{align*}(X f)(x)= x f \end{align*} $$

corresponding to multiplication with $x \in \mathbb {R}$ and its operator domain is given by

$$ \begin{align*} \mathrm{dom}(X) & = \big \{ f \in L^2_+(\mathbb{R}; \mathcal{V}) \mid x f \in L^2(\mathbb{R}; \mathcal{V}) \big \} \\ & = \big \{ f \in L^2_+(\mathbb{R}; \mathcal{V}) \mid \frac{d \widehat{f}}{d\xi} \in L^2(\mathbb{R}_+; \mathcal{V}) \text{ and } \widehat{f}(0) = 0 \big \} \,. \end{align*} $$

Note that X is the generator of the Lax–Beurling semigroup $\{ S(\eta ) \}_{\eta \geq 0}$ corresponding to right shifts on $L^2_+(\mathbb {R}; \mathcal {V})$ , that is, we have

$$ \begin{align*}(S(\eta) f)(x) = e^{\mathrm{i} \eta x} f(x) \quad \text{for } f \in L^2_+(\mathbb{R}; \mathcal{V}) \text{ and } \eta \geq 0 \,. \end{align*} $$

We will sometimes use the notation $S(\eta ) = e^{\mathrm {i} \eta X}$ . Note that the strict inclusion $\mathrm {dom}(X) \subsetneq \mathrm {dom}(X^*)$ holds, for example, the rational function $\frac {1}{x+\mathrm {i}} \in \mathrm {dom}(X^*)$ does not belong to $\mathrm {dom}(X)$ . Further details on the generators $X^*$ and X can be found in [Reference Gérard and Pushnitski17] in the scalar-valued case when $\mathcal {V}$ is replaced by $\mathbb {C}$ , but the necessary adaptations to our setting are elementary.

In addition to the generator $X^*$ , another important operator in our analysis is given by the (unbounded) linear operator

$$ \begin{align*}I_+ : \mathrm{dom}(X^*) \subset L^2_+(\mathbb{R}; \mathcal{V}) \to \mathcal{V}, \quad f \mapsto I_+(f):= \widehat{f}(0^+) = \lim_{\xi \to 0^+} \widehat{f}(\xi) \,. \end{align*} $$

Note that the definition of $I_+$ as the one-sided limit of $\widehat {f}(\xi )$ as $\xi \to 0^+$ makes sense for any $f \in \mathrm {dom}(X^*)$ by the standard trace theorem for Sobolev functions in $H^1(\mathbb {R}_+)$ . An alternative and useful expression for the action of $I_+$ is found by using the approximate identity $\chi _\varepsilon $ with

$$ \begin{align*}\chi_\varepsilon(x) := \frac{1}{1- \mathrm{i} \varepsilon x} \in L^2_+(\mathbb{R}; \mathbb{C}) \quad \text{for } \varepsilon> 0 \,. \end{align*} $$

Let $\mathbf {v} \in \mathcal {V}$ be a fixed vector. By Plancherel’s identity, we have

$$ \begin{align*}\lim_{\varepsilon \to 0} \langle f, \mathbf{v} \chi_\varepsilon \rangle = \lim_{\varepsilon \to 0} \frac{1}{\varepsilon} \int_0^\infty \langle \widehat{f}(\xi), \mathbf{v} \rangle_{\mathcal{V}} \, \mathrm{e}^{-\xi/\varepsilon} \, d \xi = \langle \widehat{f}(0^+), \mathbf{v} \rangle_{\mathcal{V}} = \langle I_+(f), \mathbf{v} \rangle_{\mathcal{V}} \,. \end{align*} $$

Thus, for any orthonormal basis $(\mathbf {v}_1, \ldots , \mathbf {v}_N)$ in $\mathcal {V}$ with $N = \dim \mathcal {V}$ , we obtain

(2.2) $$ \begin{align} I_+(f) = \lim_{\varepsilon \to 0} \sum_{k=1}^{N} \langle f, \mathbf{v}_k \chi_\varepsilon \rangle \mathbf{v}_k \,. \end{align} $$

For later use, we also record the following formula

(2.3) $$ \begin{align} \mathrm{Im} \langle X^* f, f \rangle = -\frac{1}{4 \pi} | I_+(f)|^2_{\mathcal{V}} \quad \text{for } f \in \mathrm{dom}(X^*) \, , \end{align} $$

which is a simple consequence from Plancherel’s identity and integration by parts.

Finally, we record another elementary fact involving the operators $I_+$ and $X^*$ as follows. Let $f \in L^2_+(\mathbb {R}, \mathcal {V})$ be given. As before, we suppose that $(\mathbf {v}_1, \ldots , \mathbf {v}_N)$ is an orthonormal basis of $\mathcal {V}$ with $N = \dim \mathcal {V}$ . Thus we can write

$$ \begin{align*}f(x) = \sum_{k=1}^N f_k(x) \mathbf{v}_k \end{align*} $$

with $f_k(x) = \langle f(x), \mathbf {v}_k \rangle _{\mathcal {V}} \in L^2_+(\mathbb {R}; \mathbb {C})$ for $k=1, \ldots , N$ . Since $\widehat {f}(\xi )= \sum _{k=1}^N \widehat {f}_k(\xi ) \mathbf {v}_k$ , we find

$$ \begin{align*} \widehat{f}(\xi) = \sum_{k=1}^N \lim_{\varepsilon \to 0} \left ( \int_{\mathbb{R}} \mathrm{e}^{-\mathrm{i} x \xi} \frac{ f_k(x) }{1+\mathrm{i} \varepsilon x} \, dx \right ) \mathbf{v}_k = \sum_{k=1}^N \lim_{\varepsilon \to 0} \langle S(\xi)^* f, \mathbf{v}_k\chi_\varepsilon \rangle \, \mathbf{v}_k \,. \end{align*} $$

By taking the inverse Fourier transform, we obtain, for any $z \in \mathbb {C}_+$ , that

$$ \begin{align*} f(z) & = \frac{1}{2 \pi} \int_0^{\infty} \mathrm{e}^{\mathrm{i} z \xi} \left ( \sum_{k=1}^N \lim_{\varepsilon \to 0} \langle S(\xi)^* f, \mathbf{v}_k\chi_\varepsilon \rangle \, \mathbf{v}_k \right ) d \xi \\ & = \frac{1}{2 \pi} \lim_{\varepsilon \to 0} \sum_{k=1}^N \left ( \int_0^{\infty} \langle e^{\mathrm{i} z \xi -\mathrm{i} \xi X^*} f, \mathbf{v}_k\chi_\varepsilon \rangle \, d \xi \right ) \mathbf{v}_k \\ & = \frac{1}{2 \pi \mathrm{i}} \lim_{\varepsilon \to 0} \sum_{k=1}^N \left \langle (X^*- z \mathrm{Id})^{-1} f, \mathbf{v}_k \chi_\varepsilon \right \rangle \mathbf{v}_k \,. \end{align*} $$

In view of (2.2), we therefore deduce the identity

(2.4) $$ \begin{align} \boxed{f(z) = \frac{1}{2 \pi \mathrm{i}} I_+[(X^*- z \mathrm{Id})^{-1} f] } \end{align} $$

which is valid for any $f \in L^2_+(\mathbb {R}; \mathcal {V})$ and $z \in \mathbb {C}_+$ .

Fredholm property and invariant subspaces

Recall that

$$ \begin{align*}H_{\mathbf{U}}: L^2_+(\mathbb{R}; \mathcal{V}) \to L^2_-(\mathbb{R}; \mathcal{V}), \quad H_{\mathbf{U}} f= \Pi_-(\mathbf{U} f) \end{align*} $$

denotes the corresponding (block) Hankel operator with matrix-valued symbol $\mathbf {U}$ . For later use, we remark that the adjoint Hankel operator is given by

$$ \begin{align*}H_{\mathbf{U}}^* : L^2_-(\mathbb{R}; \mathcal{V}) \to L^2_+(\mathbb{R};\mathcal{V}), \quad H_{\mathbf{U}}^* f = \Pi_+(\mathbf{U} f) \,. \end{align*} $$

We have the following general fact, for matrix-valued symbols $\mathbf {U}$ satisfying the assumptions stated above.

Lemma 4.1 (Key Identity and Fredholmness).

We have the identity

$$ \begin{align*}T_{\mathbf{U}}^2 = \mathrm{Id} - K_{\mathbf{U}} \quad \text{on } L^2_+(\mathbb{R}; \mathcal{V}) \, , \end{align*} $$

where the self-adjoint operator

$$ \begin{align*}K_{\mathbf{U}} := H^*_{\mathbf{U}} H_{\mathbf{U}} : L^2_+(\mathbb{R}; \mathcal{V}) \to L^2_+(\mathbb{R}; \mathcal{V}) \end{align*} $$

satisfies $0 \leq K_{\mathbf {U}} \leq \mathrm {Id}$ and it is trace-class with

$$ \begin{align*}\mathrm{Tr}(K_{\mathbf{U}}) = \mathrm{Tr}(H^*_{\mathbf{U}} H_{\mathbf{U}}) = \mathrm{const.} \cdot \| \mathbf{U} \|_{\dot{H}^{\frac 1 2}}^2 \,. \end{align*} $$

Moreover, the Toeplitz operator $T_{\mathbf {U}}$ is Fredholm with index 0.

Proof. Let us consider the case $\mathcal {V}= \mathbb {C}^d$ , where we remark that the proof for $\mathcal {V} = M_d(\mathbb {C})$ is analogous. Suppose that $f \in L^2_+(\mathbb {R}; \mathbb {C}^d)$ is given. Using that holds on $L^2_+(\mathbb {R}; \mathbb {C}^d)$ and $\mathbf {U}^*(x)=\mathbf {U}(x)$ for a. e. $x \in \mathbb {R}$ , we observe that

$$ \begin{align*} T_{\mathbf{U}} ( T_{\mathbf{U}} f) & = \Pi_+(\mathbf{U} \Pi_+(\mathbf{U} f)) = \Pi_+ ({\mathbf{U}} (\mathrm{Id} - \Pi_-) \mathbf{U} f) \\ & = \Pi_+ f - \Pi_+ ( {\mathbf{U}} (\Pi_- \mathbf{U} f) ) = f - H_{\mathbf{U}}^* H_{\mathbf{U}} f, \end{align*} $$

since we trivially have that $\Pi _+ f = f$ for $f \in L^2_+(\mathbb {R}; \mathbb {C}^d)$ . This proves the claimed identity.

Consider now the bounded and self-adjoint operator

$$ \begin{align*}K_{\mathbf{U}} := H_{\mathbf{U}}^* H_{\mathbf{U}} : L^2_+(\mathbb{R}; \mathbb{C}^d) \to L^2_+(\mathbb{R}; \mathbb{C}^d) \,. \end{align*} $$

Clearly, we have that $K_{\mathbf {U}} \geq 0$ is non-negative. Also, we notice that $\| K_{\mathbf {U}} \| \leq \| H_{\mathbf {U}} \|^2 \leq \| \mathbf {U} \|_{L^\infty } = 1$ , which shows that $K_{\mathbf {U}} \leq \mathrm {Id}$ holds in the sense of operators. Next, we observe that $K_{\mathbf {U}}$ is trace-class with

(4.1) $$ \begin{align} \mathrm{Tr}(K_{\mathbf{U}}) = \mathrm{Tr}(H_{\mathbf{U}}^* H_{\mathbf{U}}) = \| H_{\mathbf{U}} \|_{HS}^2 = c \cdot \| \mathbf{U} \|_{\dot{H}^{\frac 1 2}}^2 \, , \end{align} $$

where $c>0$ is some numerical constant. Here $\| A \|_{HS}$ denotes the Hilbert–Schmidt norm of a bounded operator $A : H_1 \to H_2$ with separable Hilbert spaces $H_1, H_2$ , that is, we have

$$ \begin{align*}\| A \|_{HS}^2 = \sum_{n=1}^\infty \langle A e_n, A e_n \rangle_{H_2} \, , \end{align*} $$

where $(e_n)_{n \in \mathbb {N}}$ is an arbitrary orthonormal basis of $H_1$ . For the last equation in (4.1), we give an elementary proof taken from [Reference Gérard and Lenzmann14]. Using the orthogonal decomposition $L^2(\mathbb {R};\mathbb {C}^d) = L^2_+(\mathbb {R}; \mathbb {C}^d) \oplus L^2_-(\mathbb {R}; \mathbb {C}^d)$ , we consider the commutator of $\mathbf {U}$ viewed as multiplication operator on $L^2(\mathbb {T}; \mathbb {C}^d)$ with the Hilbert transform $\mathsf {H}$ . This can be written as a $2 \times 2$ -matrix of operators such that

$$ \begin{align*}[\mathsf{H}, \mathbf{U}] = \left ( \begin{array}{cc} 0 & -\mathrm{i} \Pi_+ \mathbf{U} \Pi_- \\ \mathrm{i} \Pi_- \mathbf{U} \Pi_+ & 0 \end{array} \right ) : L^2_+(\mathbb{R}; \mathbb{C}^d) \oplus L^2_-(\mathbb{R}; \mathbb{C}^d) \to L^2_+(\mathbb{R}; \mathbb{C}^d) \oplus L^2_-(\mathbb{R};\mathbb{C}^d). \end{align*} $$

On the other hand, from the singular integral formula for $\mathsf {H}$ , we easily see that $[\mathsf {H}, \mathbf {U}]$ has the integral kernel $h_{\mathbf {U}}(x,y) = \frac {1}{\pi } \frac {\mathbf {U}(x)-\mathbf {U}(y)}{x-y} \in L^2(\mathbb {R} \times \mathbb {R}; M_d(\mathbb {C}))$ . Hence its Hilbert-Schmidt norm as an operator acting on $L^2(\mathbb {R}; \mathcal {V})$ can be directly computed as

$$ \begin{align*}\| [\mathsf{H}, \mathbf{U}] \|_{HS}^2 = \| h_{\mathbf{U}} \|_{L^2(\mathbb{R} \times \mathbb{R}; M_d(\mathbb{C}))}^2 = \frac{1}{\pi^2} \int_{\mathbb{R}} \! \int_{\mathbb{R}} \frac{|\mathbf{U}(x)- \mathbf{U}(y)|_F^2}{|x-y|^2} \, dx \,dy \, , \end{align*} $$

where $| \cdot |_F$ denotes the Frobenius norm of matrices in $M_d(\mathbb {C})$ . Next, by using that $\mathbf {U}= \mathbf {U}^*$ holds, we see that $\| h_{\mathbf {U}} \|_{HS}^2 = \| \Pi _+ \mathbf {U} \Pi _- \|_{HS}^2 + \| \Pi _- \mathbf {U} \Pi _+ \|_{HS}^2 =2 \| H_{\mathbf {U}} \|_{HS}^2$ . Recalling the formula (2.1), we deduce that the last equation in (4.1) holds.

It remains to prove that $T_{\mathbf {U}}$ is Fredholm with index 0. Indeed, we readily see that $T_{\mathbf {U}}$ is Fredholm since $T_{\mathbf {U}}$ is invertible modulo compact operators, which directly follows from the identity $T_{\mathbf {U}} T_{\mathbf {U}} = T_{\mathbf {U}}^2 = \mathrm {Id} - K_{\mathbf {U}}$ . Since $T_{\mathbf {U}}$ is self-adjoint, its Fredholm index must be 0.

In view of the general identity established in Lemma 4.1, it is natural to introduce the following closed subspaces

(4.2) $$ \begin{align} \boxed {\mathfrak{H}_0 := \mathrm{ker}(K_{\mathbf{U}}) \quad \text{and} \quad \mathfrak{H}_1 := \overline{\mathrm{ran}(K_{\mathbf{U}})} } \end{align} $$

which yields the orthogonal decomposition

(4.3) $$ \begin{align} \boxed{L^2_+(\mathbb{R}; \mathcal{V}) = \mathfrak{H}_0 \oplus \mathfrak{H}_1 } \end{align} $$

As a direct consequence, we obtain a decomposition of the Toeplitz operator $T_{\mathbf {U}}$ into invariant subspaces in the spirit of the celebrated Sz.–Nagy–Foiaș decomposition for contractions on Hilbert spaces [Reference Sz.-Nagy, Foias, Bercovici and Kérchy34] (also referred to as Langer’s lemma in [Reference Langer24]), which in turn is a generalization of the well-known Wold decomposition for isometries on Hilbert spaces. Note that $T_{\mathbf {U}}$ is a contraction, because its operator norm satisfies $\|T_{\mathbf {U}}\| \leq \| \mathbf {U} \|_{L^\infty } = 1$ , where in fact we have equality in view of the identity in Lemma 4.1 above.

Proof. Since the operator $K_{\mathbf {U}} : L^2_+(\mathbb {R}; \mathcal {V}) \to L^2_+(\mathbb {R}; \mathcal {V})$ with $0 \leq K_{\mathbf {U}} \leq 1$ is compact and self-adjoint, we can write

$$ \begin{align*}K_{\mathbf{U}} = \sum_{j=1}^N \lambda_j \langle \cdot, \varphi_j\rangle \varphi_j \end{align*} $$

with $N = \mathrm {rank}(K_{\mathbf {U}}) \in \mathbb {N} \cup \{\infty \}$ , $\lambda _j \in (0, 1]$ for $j=1, \ldots , N$ , and the corresponding eigenvectors $(\varphi _j)_{j=1}^N$ form an orthonormal basis of $\mathfrak {H}_1 = \overline {\mathrm {ran}(K_{\mathbf {U}})}$ . By the identity $T_{\mathbf {U}}^2 = \mathrm {Id} - K_{\mathbf {U}}$ and from elementary spectral calculus for the self-adjoint restrictions $T_{\mathbf {U}} |_{E_{\lambda _j}}$ on the finite-dimensional subspaces $E_{\lambda _j}=\mathrm {ker}(K_{\mathbf {U}}-\lambda _j \mathrm {Id}) \subset \mathfrak {H}_1$ , we deduce that

(4.4) $$ \begin{align} T_{\mathbf{U}} = \sum_{j=1}^N \varepsilon_j \sqrt{1- \lambda_j} \langle \cdot, \varphi_j \rangle \varphi_j \quad \text{on } \mathfrak{H}_1 \end{align} $$

with some $\varepsilon _j \in \{ \pm 1 \}$ for $j=1, \ldots , N$ . Evidently, we have that $\mathfrak {H}_1$ is invariant under $T_{\mathbf {U}}$ and we see that $T_{\mathbf {U}} |_{\mathfrak {H}_1}$ is c. n. u. Because otherwise $T_{\mathbf {U}} |_{\mathfrak {H}_1}$ would have an eigenvalue $\mu \in \{ \pm 1\}$ on some finite-dimensional subspace $E_{\lambda _j}$ , contradicting the above explicit formula since $\lambda _j> 0$ holds.

Since $\mathfrak {H}_0 = \mathrm {ker}(K_{\mathbf {U}}) = \mathfrak {H}_1^\perp $ and by self-adjointness of $T_{\mathbf {U}}$ , we see that $T_{\mathbf {U}}(\mathfrak {H}_0) \subset \mathfrak {H}_0$ . Furthermore from $T_{\mathbf {U}}^2 = \mathrm {Id} - K_{\mathbf {U}}$ , we readily find $T^2_{\mathbf {U}} |_{\mathfrak {H}_0} = \mathrm {Id} |_{\mathfrak {H}_0}$ , which implies that the self-adjoint operator $T_{\mathbf {U}} |_{\mathfrak {H}_0}$ is also unitary.

Thanks to the formula $T_{\mathbf {U}}^2 = \mathrm {Id} - K_{\mathbf {U}}$ and the decomposition obtained in Proposition 4.1, we deduce that the spectrum of $T_{\mathbf {U}}$ decomposes as

$$ \begin{align*}\sigma(T_{\mathbf{U}}) = \sigma_{\mathrm{e}}(T_{\mathbf{U}}) \sqcup \sigma_{\mathrm{d}}(T_{\mathbf{U}}) \, , \end{align*} $$

where the essential and discrete spectra of $T_{\mathbf {U}}$ are given by

$$ \begin{align*}\sigma_{\mathrm{e}}(T_{\mathbf{U}}) = \sigma(T_{\mathbf{U}} |_{\mathfrak{H}_0}) \subset \{ \pm 1 \} \end{align*} $$

$$ \begin{align*}\sigma_{\mathrm{d}}(T_{\mathbf{U}}) = \sigma_{\mathrm{d}}( T_{\mathbf{U}} |_{\mathfrak{H}_1}) = \{ \varepsilon_j \sqrt{1-\lambda_j} \mid j=1,\ldots, \mathrm{rank}(K_{\mathbf{U}}) \} \, , \end{align*} $$

with the sequences $(\varepsilon _j)\subset \{ \pm 1 \}$ and $(\lambda _j) \subset (0, 1]$ taken from (4.4) above.

As a next step, we find some explicit elements in the invariant subspace $\mathfrak {H}_1$ . The use of this fact will become clear later on when proving our global well-posedness result for (HWM $_d$ ). Recall our assumption that

(4.5) $$ \begin{align} \mathbf{U} = \mathbf{U}_\infty + \mathbf{V} \in M_d(\mathbb{C}) \oplus H^{\frac 1 2}(\mathbb{R}; M_d(\mathbb{C})) \,. \end{align} $$

For later use, we make the following observation. For better readability, we use $B.A$ to denote the product $BA$ of two matrices $B, A\in M_d(\mathbb {C})$ .

Proof. Since $K_{\mathbf {U}} = H^*_{\mathbf {U}} H_{\mathbf {U}}$ , we find that $\mathrm {ker}(K_{\mathbf {U}}) = \mathrm {ker}(H_{\mathbf {U}})$ , which yields that $\mathfrak {H}_1= (\mathfrak {H}_0)^\perp = \overline {\mathrm {ran}(H^*_{\mathbf {U}})}$ . Hence we have to show that $\Pi _+ \mathbf {V}.A \in \overline {\mathrm {ran}(H^*_{\mathbf {U}})}$ holds for any constant matrix $A \in M_d(\mathbb {C})$ . Indeed, by recalling $\chi _\varepsilon = \frac {1}{1-\mathrm {i} \varepsilon x} \in L^2_+(\mathbb {R}; \mathbb {C})$ for $\varepsilon> 0$ and hence $\overline {\chi }_\varepsilon \in L^2_-(\mathbb {R}; \mathbb {C})$ , we notice

$$ \begin{align*} \lim_{\varepsilon \to 0} H^*_{\mathbf{U}}(\overline{\chi}_\varepsilon A) & = \lim_{\varepsilon \to 0} \Pi_+( \mathbf{U}. \overline{\chi}_\varepsilon A) \\ & = \lim_{\varepsilon \to 0} \Pi_+ ( (\mathbf{U}_\infty + \Pi_+ \mathbf{V} + \Pi_- \mathbf{V}). \overline{\chi}_\varepsilon A ) \\ & = \lim_{\varepsilon \to 0} \Pi_+( (\Pi_+\mathbf{V}) \overline{\chi_\varepsilon}).A = \Pi_+ \mathbf{V}.A, \end{align*} $$

because of $\lim _{\varepsilon \to 0} \Pi _+ ( f \overline {\chi }_\varepsilon ) = f$ in $L^2_+(\mathbb {R}; \mathcal {V})$ by dominated convergence. This shows that $\Pi _+ \mathbf {V}.A$ belongs to $\mathfrak {H}_1=\overline {\mathrm {ran}(H^*_{\mathbf {U}})}$ .

Next, by using the well-known fact that kernels of Hankel operators are invariant under the Lax–Beurling semigroup $\{ S(\eta ) \}_{\eta \geq 0}$ , we obtain the following result.

Proof. Let $f \in \mathfrak {H}_0 = \mathrm {ker}(K_{\mathbf {U}}) = \mathrm {ker}(H_{\mathbf {U}})$ . For any $\eta \geq 0$ , we immediately observe that

$$ \begin{align*} H_{\mathbf{U}}(S(\eta) f) & = \Pi_-(\mathbf{U} S(\eta) f) = \Pi_- (\mathrm{e}^{\mathrm{i} \eta x} \mathbf{U} f) \\ & = \Pi_- (\mathrm{e}^{\mathrm{i} \eta x} \Pi_-(\mathbf{U} f)) = \Pi_- (\mathrm{e}^{\mathrm{i} \eta x} H_{\mathbf{U}}(\mathbf{U} f)) = 0 \,. \end{align*} $$

Thus we find $S(\eta ) f \in \mathfrak {H}_0$ for any $f \in \mathfrak {H}_0$ . This proves that $S(\eta ) \mathfrak {H}_0 \subset \mathfrak {H}_0$ .

Since $\mathfrak {H}_0 \perp \mathfrak {H}_1$ , we directly see that $S(\eta )^* f \in \mathfrak {H}_1$ for any $f \in \mathfrak {H}_1$ and $\eta \geq 0$ with the adjoint Lax–Beurling semigroup $\{ S(\eta )^* \}_{\eta \geq 0}$ acting on $L^2_+(\mathbb {R}; \mathcal {V})$ .

Remark. As a direct consequence of the well-known Lax–Beurling theorem (see the version in [Reference Lax25] for a direct application to our setting) about invariant subspaces of $S(\eta )$ , we can deduce the following fact: If $\mathfrak {H}_0 = \mathrm {ker}(K_{\mathbf {U}}) \neq \{ 0 \}$ is nontrivial, there exist a subspace $\mathcal {V}' \subseteq \mathcal {V}$ and a function

$$ \begin{align*}\Theta \in L^\infty_+(\mathbb{R}; \mathrm{End}(\mathcal{V}'; \mathcal{V})) \quad \text{with} \quad \Theta(x)^* \Theta(x) = \mathrm{Id}_{\mathcal{V}'} \quad \text{for a. e. } x \in \mathbb{R} \end{align*} $$

such that

$$ \begin{align*}\mathfrak{H}_0 = \Theta L^2_+(\mathbb{R}; \mathcal{V}') \quad \text{and} \quad \mathfrak{H}_1 = (\Theta L^2_+(\mathbb{R}; \mathcal{V}'))^\perp \,. \end{align*} $$

The matrix-valued function $\Theta $ is called a (left) inner function and the subspace $\mathfrak {H}_1$ is thus the model space generated by $\Theta $ . However, we will not exploit this fact in the present paper.

Spectral properties for rational data

Recall that $\mathfrak {H}_1= \overline {\mathrm {ran}(K_{\mathbf {U}})}$ . We have the following characterization when the subspace $\mathfrak {H}_1$ is finite-dimensional, corresponding to the fact that the compact operator $K_{\mathbf {U}} : L^2_+(\mathbb {R}; \mathcal {V}) \to L^2_+(\mathbb {R}; \mathcal {V})$ has finite rank.

Proof. Since $\dim \mathrm {ran}(H_{\mathbf {U}}^* H_{\mathbf {U}}) = \dim \mathrm {ran}(H_{\mathbf {U}})$ , it suffices to consider the Hankel operator $H_{\mathbf {U}} : L^2_+(\mathbb {R}; \mathcal {V}) \to L^2_-(\mathbb {R}; \mathcal {V})$ . Furthermore, since $\mathbf {U}_\infty \in M_d(\mathbb {C})$ is constant, we see that $H_{\mathbf {U}_\infty }=0$ and hence $H_{\mathbf {U}}= H_{\mathbf {U}_\infty + \mathbf {V}} = H_{\mathbf {V}}$ . Thus it remains to discuss $H_{\mathbf {V}}$ with the matrix-valued symbol $\mathbf {V} \in (H^{\frac 1 2} \cap L^\infty )(\mathbb {R}; M_d(\mathbb {C}))$ for the rest of the proof.

We first recall the following general Kronecker-type theorem valid for Hankel operators acting on the Hardy space $L_+^2(\mathbb {T}; \mathcal {H})$ on the torus $\mathbb {T} \cong \partial \mathbb {D}$ , where $\mathcal {H}$ is a given separable complex Hilbert space (not necessarily finite-dimensional). Correspondingly, we use $\mathbb {P}_+$ and $\mathbb {P}_- = \mathrm {Id} - \mathbb {P}_+$ to denote the Cauchy–Szegő projections on $L^2(\mathbb {T}; \mathcal {E})$ ; see [Reference Peller32] for a general background. As usual, we use $\mathcal {B}(\mathcal {H}, \mathcal {K})$ to denote the Banach space of bounded linear operators from $\mathcal {H}$ to another complex Hilbert space $\mathcal {K}$ . From [Reference Peller32][Chapter 2, Theorem 5.3] we directly deduce the following result.

Theorem 4.1 (Kronecker’s theorem on $L^2_+(\mathbb {T}; \mathcal {H})$ ).

Let $\mathcal {H}, \mathcal {K}$ be separable complex Hilbert spaces and assume $\Phi \in L^\infty (\mathbb {T}; \mathcal {B}(\mathcal {H}, \mathcal {K}))$ . Define the Hankel operator $H_\Phi : L^2_+(\mathbb {T}; \mathcal {H}) \to L^2_-(\mathbb {T}; \mathcal {K})$ by $H_\Phi f = \mathbb {P}_-(\Phi f)$ . Then $\mathrm {rank} \, H_\Phi < \infty $ if and only if $\mathbb {P}_- \Phi : \mathbb {T} \to \mathcal {B}(\mathcal {H}, \mathcal {K})$ is a rational map of the form

$$ \begin{align*}\mathbb{P}_- \Phi = \sum_{\lambda \in \Lambda} \sum_{n=1}^{k(\lambda)} \frac{T_{\lambda, n}}{(z- \lambda)^n} \, , \end{align*} $$

where $\Lambda $ is a finite subset in $\mathbb {D}$ and the $k(\lambda )$ are positive integers and $T_{n, \lambda } \in \mathcal {B}(\mathcal {H}, \mathcal {K}) \setminus \{ 0 \}$ are finite-rank operators, $\lambda \in \Lambda $ , and $1 \leq n \leq k(\lambda )$ .

Let us now take the finite-dimensional spaces $\mathcal {H}=\mathcal {K} = \mathcal {V}$ in the previous result with either $\mathcal {V} = \mathbb {C}^d$ or $\mathcal {V} = M_d(\mathbb {C})$ . For any $\Phi \in L^\infty (\mathbb {T}; M_d(\mathbb {C}))$ given, we deduce the equivalence

$$ \begin{align*}H_{\Phi} = \mathbb{P}_- \Phi \mathbb{P}_+ \text{ has finite rank if and only if } \mathbb{P}_- \Phi \in L^\infty(\mathbb{T}; M_d(\mathbb{C})) \text{ is rational.} \end{align*} $$

Now using the standard conformal map $\omega : \mathbb {D} \to \mathbb {C}_+$ with $\omega (\zeta ) = \mathrm {i} \frac {1 + \zeta }{1-\zeta }$ , let us define the map

$$ \begin{align*}(\mathcal{U} f)(x) = \frac{1}{\sqrt{\pi}} \frac{(f \circ \omega^{-1})(x)}{x+\mathrm{i}} = \frac{1}{\sqrt{\pi}} \frac{1}{x+ \mathrm{i}} f \left ( \frac{x-\mathrm{i}}{x+\mathrm{i}} \right ) \quad \text{for } f \in L^2(\mathbb{T}; \mathcal{V}), \end{align*} $$

which is known to be unitary operator from $L^2(\mathbb {T}; \mathcal {V})$ to $L^2(\mathbb {R}; \mathcal {V})$ with the property that $\mathcal {U}(L^2_+(\mathbb {T}; \mathcal {V})) = L^2_+(\mathbb {R}; \mathcal {V})$ ; see [Reference Peller32][Appendix 2.1]. We easily verify that

$$ \begin{align*}H_{\Phi} = \mathcal{U}^* H_{\mathbf{V}} \mathcal{U} \quad \text{with } \Phi = \mathbf{V} \circ \omega, \end{align*} $$

see, for example, [Reference Peller32][Chapter 1, Lemma 8.3]. Since compositions with $\omega $ and $\omega ^{-1}$ preserve rationality and in view of the identity $\Pi _- \mathbf {V} = (\mathbb {P}_- ( \mathbf {V} \circ \omega )) \circ \omega ^{-1}$ , we deduce

$$ \begin{align*}H_{\mathbf{V}} \text{ has finite rank if and only if } \Pi_- \mathbf{V} : \mathbb{R} \to M_d(\mathbb{C}) \text{ is rational.} \end{align*} $$

Finally, by recalling that $\Pi _+ \mathbf {V} = (\Pi _- \mathbf {V})^*$ , we conclude that $\mathbf {V} = (\Pi _- \mathbf {V})^* + \Pi _- \mathbf {V}$ is a rational function if and only if $H_{\mathbf {V}}$ has finite rank.

We now show that, for rational matrix-valued symbols $\mathbf {U}$ , the subspace $\mathfrak {H}_1$ is also an invariant subspace for the unbounded operator $X^*$ , which is the generator of the adjoint Lax–Beurling semigroup $\{ S(\eta )^* \}_{\eta \geq 0}=\{ \mathrm {e}^{-\mathrm {i} \eta X^*} \}_{\eta \geq 0}$ .

Local well-posedness for sufficiently regular data

We start with a result on local well-posedness for the matrix-valued (HWM $_d$ ) for sufficiently regular initial data of the form

(5.1) $$ \begin{align} \mathbf{U}_0(x) = \mathbf{U}_\infty + \mathbf{V}_0(x) \in M_d(\mathbb{C}) \oplus H^s(\mathbb{R}; M_d(\mathbb{C})) \, , \end{align} $$

satisfying the constraints

(5.2)

In what follows, we will always assume that

$$ \begin{align*}s> \frac{3}{2} \,. \end{align*} $$

In particular, the initial datum $\mathbf {U}_0 : \mathbb {R} \to M_d(\mathbb {C})$ is of class $C^1$ by Sobolev embeddings. In view of (5.2), we easily conclude that $\mathrm {Tr}(\mathbf {U}_0(x))$ can only attain integer values, whence it follows $\mathrm {Tr}(\mathbf {U}_0(x)) = \text {const}.$ on $\mathbb {R}$ by continuity.Footnote ⁴ As a consequence, we deduce that there exists some integer $0 \leq k \leq d$ such that

$$ \begin{align*}\mathbf{U}_0(x) \in \mathsf{Gr}_k(\mathbb{C}^d) \quad \text{for } x \in \mathbb{R}. \end{align*} $$

We have the following result.

Lemma 5.1. Let $s> \frac {3}{2}$ , $d \geq 2$ , and assume $\mathbf {U}_0 : \mathbb {R} \to M_d(\mathbb {C})$ satisfies (5.1) and (5.2). Then, for any $R> 0$ , there exists some $T=T(R)>0$ such that for every $\mathbf {U}_0=\mathbf {U}_\infty + \mathbf {V}_0$ as above with $\| \mathbf {V}_0 \|_{H^s} < R$ , there exists a unique solution of (HWM _d ) of the form

$$ \begin{align*}\mathbf{U}(t) = \mathbf{U}_\infty + \mathbf{V}(t) \in M_d(\mathbb{C}) \oplus C([0,T]; H^s(\mathbb{R};M_d(\mathbb{C}))) \end{align*} $$

and we have $\mathbf {U}(t,x) \in \mathsf {Gr}_k(\mathbb {C}^d)$ for all $t \in [0,T]$ and $x \in \mathbb {R}$ with some integer $0 \leq k \leq d$ .

Furthermore, the $H^\sigma $ -regularity of $\mathbf {V}_0$ with $\sigma> s$ is propagated on the whole maximal time-interval of existence of $\mathbf {U}(t)$ , and the flow map $\mathbf {V}_0 \mapsto \mathbf {V}(t)$ is continuous in the $H^\sigma $ -topology.

Explicit flow formula

Inspired by the very recent work [Reference Gérard10] on the Benjamin–Ono equation, we next derive an explicit flow formula for solutions of (HWM _d ) based on its Lax pair structure acting on the Hardy space. Note that, in this formula, we choose the vector space $\mathcal {V} = M_d(\mathbb {C})$ for the Hardy space $L^2_+(\mathbb {R}; \mathcal {V})$ .

Lemma 5.2 (Explicit Flow Formula).

Let $s> \frac {3}{2}$ , $d \geq 2$ , and $\mathbf {U}(t) = \mathbf {U}_\infty + \mathbf {V}(t) \in M_d(\mathbb {C}) \oplus C([0,T]; H^s(\mathbb {R}, M_d(\mathbb {C}))$ be as in Lemma 5.1 above. Then it holds that

$$ \begin{align*}\Pi_+ \mathbf{V}(t,z) = \frac{1}{2 \pi \mathrm{i}} I_+ \left [ (X^*+tT_{\mathbf{U}_0} - z \mathrm{Id})^{-1} \Pi_+ \mathbf{V}_0 \right ] \quad \text{for } z \in \mathbb{C}_+ \text{ and } t \in [0,T]. \end{align*} $$

Here $T_{\mathbf {U}_0} : L^2_+(\mathbb {R}; \mathcal {V}) \to L^2_+(\mathbb {R}; \mathcal {V})$ denotes the Toeplitz operator $T_{\mathbf {U}_0} f = \Pi _+(\mathbf {U}_0 f)$ with $\mathcal {V} = M_d(\mathbb {C})$ .

Before we turn to the proof of Lemma 5.2, we need some commutator identities as follows. Recall that

$$ \begin{align*}B_{\mathbf{U}} = -\frac{\mathrm{i}}{2} ( \mu_{\mathbf{U}} |D| + |D| \mu_{\mathbf{U}} ) + \frac{\mathrm{i}}{2} \mu_{|D| \mathbf{U}} \,. \end{align*} $$

In fact, since we can restrict to the Hardy space $L^2_+(\mathbb {R}; \mathcal {V})$ , it will be convenient to work with the compression of $B_{\mathbf {U}}$ to the Hardy space $L^2_+(\mathbb {R}; \mathcal {V})$ denoted by

$$ \begin{align*}B^+_{\mathbf{U}} := \Pi_+ B_{\mathbf{U}} \Pi_+ = -\frac{\mathrm{i}}{2} (T_{\mathbf{U}} D + D T_{\mathbf{U}}) + \frac{\mathrm{i}}{2} T_{|D| \mathbf{U}} \quad \text{with } D=-\mathrm{i} \partial_x \,. \end{align*} $$

Note that $D \geq 0$ on $L^2_+(\mathbb {R}; \mathcal {V})$ with its operator domain $\mathrm {dom}(D) = H^1_+(\mathbb {R}; \mathcal {V})$ . The Lax equation for $T_{\mathbf {U}(t)} : L^2_+(\mathbb {R}, \mathcal {V}) \to L^2_+(\mathbb {R}, \mathcal {V})$ can thus be written as

$$ \begin{align*}\frac{d}{dt} T_{\mathbf{U}(t)} = [B_{\mathbf{U}(t)}^+, T_{\mathbf{U}(t)} ] \,. \end{align*} $$

We have the following key commutator identity.

Proposition 5.1. For any $f \in \mathrm {dom}(X^*) \cap H^1_+(\mathbb {R}; \mathcal {V})$ , it holds that

$$ \begin{align*}[X^*, B^+_{\mathbf{U}}] f = T_{\mathbf{U}} f \,. \end{align*} $$

Proof. Using the fact that $[X^*, D] = \mathrm {i} \, \mathrm {Id}$ and by Lemma 3.2, we calculate

$$ \begin{align*} [X^*, T_{\mathbf{U}} D + D T_{\mathbf{U}}] f & = [X^*, T_{\mathbf{U}}] Df + T_{\mathbf{U}} [X^*,D] f + [X^*,D] T_{\mathbf{U}} f + D [X^*,T_{\mathbf{U}}] f \\ & = \frac{\mathrm{i}}{2 \pi} \Pi_+ \mathbf{V}.I_+(Df) + \mathrm{i} T_{\mathbf{U}} f + \mathrm{i} T_{\mathbf{U}} f + \frac{\mathrm{i}}{2 \pi} D (\Pi_+ \mathbf{V}.I_+(f)) \\ & = 2 \mathrm{i} T_{\mathbf{U}} f + \frac{\mathrm{i}}{2 \pi} \Pi_+(D \mathbf{V}).I_+(f) \, , \end{align*} $$

where also used that $I_+(Df)=0$ holds. By applying Lemma 3.2 once again,

$$ \begin{align*}[X^*,T_{|D| \mathbf{U}}] f = \frac{\mathrm{i}}{2 \pi} \Pi_+(|D| \mathbf{V}).I_+(f) = \frac{\mathrm{i}}{2 \pi} \Pi_+(D \mathbf{V}).I_+(f) \,. \end{align*} $$

In view of these identities, we easily conclude the claimed identity.

We are now ready to turn to the proof of Lemma 5.2.

Proof of Lemma 5.2 (Explicit Flow Formula).

We divide the proof into the following steps. We remind the reader that we take $\mathcal {V} = M_d(\mathbb {C})$ in the following.

Step 1. Recalling identity (2.4), we write

$$ \begin{align*}\Pi_+ \mathbf{V}(t,z) = \frac{1}{2 \pi \mathrm{i}} I_+((X^*-z \mathrm{Id})^{-1} \Pi_+\mathbf{V}(t)) \quad \text{for } z \in \mathbb{C}_+ \text{ and } t \in [0,T] \,. \end{align*} $$

Let $F \in M_d(\mathbb {C})$ and $z \in \mathbb {C}_+$ be fixed from now on. We find

$$ \begin{align*} \langle \Pi_+ \mathbf{V}(t,z), F \rangle_{\mathcal{V}} & = \frac{1}{2 \pi \mathrm{i}} \langle I_+((X^*-z \mathrm{Id})^{-1} \Pi_+ \mathbf{V}(t), F \rangle_{\mathcal{V}} \\ &= \frac{1}{2 \pi \mathrm{i}} \lim_{\varepsilon \to 0} \left \langle (X^*-z \mathrm{Id})^{-1} \Pi_+ \mathbf{V}(t), F \chi_\varepsilon \right \rangle \\ & = \frac{1}{2 \pi \mathrm{i}} \lim_{\varepsilon \to 0} \left \langle \mathcal{U}(t)^* (X^*- z \mathrm{Id})^{-1} \Pi_+ \mathbf{V}(t), \mathcal{U}(t)^* (F \chi_\varepsilon) \right \rangle \, , \end{align*} $$

where we also use that $\mathcal {U}(t)^* : L^2_+(\mathbb {R}; \mathcal {V}) \to L^2_+(\mathbb {R}; \mathcal {V})$ is a unitary map for any $t \in [0,T]$ , which is given by the solution of the initial-value problem

$$ \begin{align*}\frac{d}{dt} \mathcal{U}(t) = B_{\mathbf{U}(t)}^+ \mathcal{U}(t) \quad \text{for } t \in [0,T], \quad \mathcal{U}(0) = \mathrm{Id} \,. \end{align*} $$

See Appendix D for details. Using the identity

$$ \begin{align*}\mathcal{U}(t)^* (X^*-z \mathrm{Id})^{-1} = (\mathcal{U}(t)^* X^* \mathcal{U}(t) - z \mathrm{Id})^{-1} \mathcal{U}(t)^* \, , \end{align*} $$

we conclude

$$ \begin{align*}\langle \Pi_+ \mathbf{V}(t,z), F \rangle_{\mathcal{V}} = \frac{1}{2 \pi \mathrm{i}} \lim_{\varepsilon \to 0} \left \langle (\mathcal{U}(t)^* X^* \mathcal{U}(t) - z \mathrm{Id})^{-1} \mathcal{U}(t)^* (\Pi_+ \mathbf{V}(t)), \mathcal{U}(t)^*(F \chi_\varepsilon) \right \rangle \end{align*} $$

for any $z \in \mathbb {C}_+$ , $t \in [0,T]$ and $F \in M_d(\mathbb {C})$ .

Step 2. We will now discuss the individual terms which appear in the expression derived in Step 1 above. First, we notice that

$$ \begin{align*}\frac{d}{dt} \mathcal{U}(t)^* X^* \mathcal{U}(t) = \mathcal{U}(t)^* [X^*, B_{\mathbf{U}(t)}^+] \mathcal{U}(t) = \mathcal{U}(t)^* T_{\mathbf{U}(t)} \mathcal{U}(t) = T_{\mathbf{U}_0} \, , \end{align*} $$

where we used Proposition 5.1 together with the fact that $T_{\mathbf {U}(t)} = \mathcal {U}(t) T_{\mathbf {U}_0} \mathcal {U}(t)^*$ holds thanks to the Lax evolution. By integration on the interval $[0,t]$ , we get

(5.3) $$ \begin{align} \mathcal{U}(t)^* X^* \mathcal{U}(t) = X^* + t T_{\mathbf{U}_0} \,. \end{align} $$

Next, we observe

$$ \begin{align*}\frac{d}{dt} ( \mathcal{U}(t)^* (F \chi_\varepsilon)) = -\mathcal{U}(t)^*(B_{\mathbf{U}(t)}^+ (F \chi_\varepsilon)) = o(1) \, , \end{align*} $$

where $o(1) \to 0$ in $L^2$ as $\varepsilon \to 0$ uniformly with respect to $t \in [0,T]$ . To see this, we remark

$$ \begin{align*} B_{\mathbf{U}}^+ (F \chi_\varepsilon) & = \frac{\mathrm{i}}{2} T_{\mathbf{U}}(F D\chi_\varepsilon) + \frac{\mathrm{i}}{2} D(T_{\mathbf{U}} F \chi_\varepsilon) - \frac{\mathrm{i}}{2} T_{|D| \mathbf{U}}(F \chi_\varepsilon) \\ & \to \frac{\mathrm{i}}{2} \Pi_+(D \mathbf{U}). F -\frac{\mathrm{i}}{2} \Pi_+(D\mathbf{U}). F = 0 \quad \text{as } \varepsilon \to 0 \end{align*} $$

in $L^2(\mathbb {R}; \mathcal {V})$ uniformly in $t \in [0,T]$ . Therefore, by integrating in t, we conclude

(5.4) $$ \begin{align} \mathcal{U}(t)^*(F \chi_\varepsilon) = F \chi_\varepsilon + o(1) \end{align} $$

with $o(1) \to 0$ in $L^2_+(\mathbb {R}, \mathcal {V})$ as $\varepsilon \to 0$ uniformly in $t \in [0,T]$ .

It remains to discuss the last term from Step 1. Here we claim that

(5.5) $$ \begin{align} \mathcal{U}(t)^* (\Pi_+ \mathbf{V}(t)) = \Pi_+ \mathbf{V}_0 \,. \end{align} $$

Since $\mathcal {U}(0)^* = \mathrm {Id}$ , we need to show that the time derivative of the left-hand side vanishes. Indeed, we note

$$ \begin{align*}\frac{d}{dt} \left ( \mathcal{U}(t)^*(\Pi_+ \mathbf{V}(t)) \right ) = \mathcal{U}(t)^* \left ( -B_{\mathbf{U}(t)}^+ \Pi_+ \mathbf{V}(t) + \partial_t \Pi_+ \mathbf{V}(t) \right ) \,. \end{align*} $$

Now, by the Lax equation $\frac {d}{dt} T_{\mathbf {U}(t)} = [B_{\mathbf {U}(t)}^+, T_{\mathbf {U}(t)}]$ and if we let denote the identity matrix in $M_d(\mathbb {C})$ , we find

$$ \begin{align*}\frac{d}{dt} T_{\mathbf{U}(t)} (E \chi_\varepsilon) = B_{\mathbf{U}(t)}^+ T_{\mathbf{U}(t)} (E \chi_\varepsilon) - T_{\mathbf{U}(t)} B_{\mathbf{U}(t)}^+ (E \chi_\varepsilon) \end{align*} $$

For the first term on right-hand side, we observe

$$ \begin{align*}B_{\mathbf{U}(t)}^+ T_{\mathbf{U}(t)} (E \chi_\varepsilon) \to B_{\mathbf{U}(t)}^+ (\Pi_+ \mathbf{V}(t)) \quad \text{in } L^2_+ \text{ as } \varepsilon \to 0 \end{align*} $$

uniformly in $t \in [0,T]$ . Furthermore, in the same way as in the discussion showing that $B_{\mathbf {U}(t)}^+ (F \chi _\varepsilon ) \to 0$ as $\varepsilon \to 0$ for any constant matrix $F \in M_d(\mathbb {C})$ , we conclude

$$ \begin{align*}T_{\mathbf{U}(t)} B_{\mathbf{U}(t)}^+(E \chi_\varepsilon) \to 0 \quad \text{in } L^2_+ \text{ as } \varepsilon \to 0 \end{align*} $$

uniformly in t. On the other hand, we have

$$ \begin{align*}\frac{d}{dt} T_{\mathbf{U}}(t) E \chi_\varepsilon \to \partial_t \Pi_+ \mathbf{V} \quad \text{in } L^2_+ \text{ as } \varepsilon \to 0 \end{align*} $$

uniformly in $t \in [0,T]$ . In summary, we infer that $\partial _t \Pi _+ \mathbf {V}(t) = B_{\mathbf {U}(t)} \Pi _+ \mathbf {V}(t)$ holds, whence it follows

$$ \begin{align*}\frac{d}{dt} \left ( \mathcal{U}(t)^* (\Pi_+ \mathbf{V}(t)) \right ) = \mathcal{U}(t)^* (-B_{\mathbf{U}(t)}^+ \Pi_+ \mathbf{V}(t) + \partial_t \Pi_+ \mathbf{V}(t) ) = 0 \,. \end{align*} $$

This completes the proof of (5.5).

Step 3. Combining the results from Step 1 and Step 2 above, we conclude, for any $F \in M_d(\mathbb {C})$ and $z \in \mathbb {C}_+$ , that

$$ \begin{align*} \langle \Pi_+ \mathbf{V}(t,z), F \rangle_{\mathcal{V}} & = \frac{1}{2 \pi \mathrm{i}} \lim_{\varepsilon \to 0} \left \langle (\mathcal{U}(t)^* X^* \mathcal{U}(t) - z \mathrm{Id})^{-1} \mathcal{U}(t)^* (\Pi_+ \mathbf{V}(t)), \mathcal{U}(t)^*(F \chi_\varepsilon) \right \rangle \, \\ & = \frac{1}{2 \pi \mathrm{i}} \lim_{\varepsilon \to 0} \left \langle (X^*+ t T_{\mathbf{U}_0} - z \mathrm{Id})^{-1} \Pi_+ \mathbf{V}_0, F \chi_\varepsilon \right \rangle \\ & = \frac{1}{2 \pi \mathrm{i}}\left \langle I_+[(X^*+tT_{\mathbf{U}_0} - z \mathrm{Id})^{-1} \Pi_+ \mathbf{V}_0)], F \right \rangle_{\mathcal{V}} \,. \end{align*} $$

Since $F \in M_d(\mathbb {C})$ is arbitrary, we deduce the claimed formula for $\Pi _+ \mathbf {V}(t,z) \in \mathcal {V}$ .

The proof of Lemma 5.2 is now complete.

Proof of Theorem 1.5

We are now ready to prove global well-posedness for (HWM $_d$ ) for rational initial data

$$ \begin{align*}\mathbf{U}_0 \in \mathcal{R}at(\mathbb{R}; \mathsf{Gr}_k(\mathbb{C}^d)) \end{align*} $$

where $d \geq 2$ and $0 \leq k \leq d$ are given integers. We note that

$$ \begin{align*}\mathbf{U}_0 = \mathbf{U}_\infty + \mathbf{V}_0 \in \mathsf{Gr}_k(\mathbb{C}^d) \oplus H^\infty(\mathbb{R}; M_d(\mathbb{C})) \end{align*} $$

holds. Hence, by the local well-posedness result from Lemma 5.1, there exists a unique maximal solution

$$ \begin{align*}\mathbf{U}(t) = \mathbf{U}_\infty + \mathbf{V}(t) \in C([0,T_{\max}); \mathsf{Gr}_k(\mathbb{C}^d) \oplus H^\infty(\mathbb{R}; M_d(\mathbb{C})) \end{align*} $$

of (HWM $_d$ ) with initial datum $\mathbf {U}(0) = \mathbf {U}_0$ and maximal (forward) time of existence $T_{\max } \in (0, +\infty ]$ such that the following implication holds:

(6.1) $$ \begin{align} T_{\max} < +\infty \quad \Rightarrow \quad \lim_{t \nearrow T_{\max}} \| \mathbf{V}(t) \|_{H^2} = +\infty \,. \end{align} $$

Thus to show that $T_{\max } = +\infty $ holds true we argue by contradiction and we suppose that $T_{\max } < +\infty $ . We now claim that

(6.2) $$ \begin{align} \sup_{t \in [0,T_{\max})} \| \mathbf{V}(t) \|_{H^2} < +\infty \, , \end{align} $$

which implies that $T_{\max } = +\infty $ must hold by (6.1). To prove (6.2), we first note that $\mathbf {V}(t,x)=\mathbf {V}(t,x)^*$ for $t \in [0,T_{\max })$ and $x \in \mathbb {R}$ . Therefore $\mathbf {V}(t) = \Pi _+ \mathbf {V}(t) + (\Pi _+ \mathbf {V}(t))^*$ and hence it suffices to show that

(6.3) $$ \begin{align} \sup_{t \in [0,T_{\max})} \| \Pi_+ \mathbf{V}(t) \|_{H^2} < +\infty \,. \end{align} $$

In view of the explicit flow formula in Lemma 5.2, we define

(6.4) $$ \begin{align} \mathrm{EF}[\mathbf{U}_0](t,z) := \frac{1}{2 \pi \mathrm{i}} I_+ \left [( X^*+ t T_{\mathbf{U}_0} -z \mathrm{Id})^{-1} \Pi_+ \mathbf{V}_0 \right ] \quad \text{for } t \geq 0 \text{ and } z \in \overline{\mathbb{C}}_+ \, , \end{align} $$

where $T_{\mathbf {U}_0} : L^2_+(\mathbb {R}; M_d(\mathbb {C})) \to L^2_+(\mathbb {R}; M_d(\mathbb {C}))$ . Let us check that $\mathrm {EF}[\mathbf {U}_0]$ is indeed well-defined for all $t \geq 0$ and $z \in \overline {\mathbb {C}}_+$ . By Lemma 4.3 (Kronecker-type theorem), the subspace $\mathfrak {H}_1 = \overline {\mathrm {ran}(K_{\mathbf {U}_0})} \subset \mathrm {dom}(X^*)$ is finite-dimensional. By Propositions 4.2 and 4.3, we deduce $ \Pi _+ \mathbf {V}_0 \in \mathfrak {H}_1$ and that

$$ \begin{align*}M(t) := X^* + t T_{\mathbf{U}_0} : \mathfrak{H}_1 \to \mathfrak{H}_1 \end{align*} $$

is an endomorphism on the finite-dimensional subspace $\mathfrak {H}_1$ . Moreover, by Lemma 6.1 with $\mathbf {W} = t \mathbf {U}_0$ , we see that the eigenvalues of $M(t)$ cannot be real, that is, $\sigma (M(t)) \cap \mathbb {R} = \emptyset $ holds, which implies that

$$ \begin{align*}\sigma(M(t)) \subset \mathbb{C}_- \quad \text{for all } t \geq 0 \,. \end{align*} $$

Hence the resolvent $(X^* + t T_{\mathbf {U}_0}- z \mathrm {Id})^{-1} : \mathfrak {H}_1 \to \mathfrak {H}_1$ exists for all $t \geq 0$ and $z \in \overline {\mathbb {C}}_+$ . Moreover, by continuity of eigenvalues of $M(t)$ with respect to t, we deduce that, for any compact interval $I \subset [0,\infty )$ , it holds that

$$ \begin{align*}\| (X^* + t T_{\mathbf{U}_0}-z\mathrm{Id})^{-1} \|_{\mathfrak{H}_1 \to \mathfrak{H}_1} \leq C(I, \mathbf{U}_0) \quad \text{for all } t \in I \text{ and } z \in \overline{\mathbb{C}}_+ \, , \end{align*} $$

with some finite constant $C(I, \mathbf {U}_0)>0$ . Since $I_+ : \mathfrak {H}_1 \subset \mathrm {dom}(X^*) \to M_d(\mathbb {C})$ is bounded (as a linear map on a finite-dimensional Hilbert space), we deduce $\mathrm {EF}[\mathbf {U}_0](t,z)$ is a rational function in z for any $t \geq 0$ , whose poles belong to a compact subset $K =K(I, \mathbf {U}_0) \subset \mathbb {C}_-$ when $t \in I$ for any given compact time interval $I \subset [0,\infty )$ .

To summarize, we have shown that, for any given compact interval $I \subset [0,\infty )$ , there exists some constant $C = C(I, \mathbf {U}_0)> 0$ such that

$$ \begin{align*}|\alpha | + \frac{1}{|\mathrm{Im} \, \alpha|} \leq C \end{align*} $$

whenever $\alpha $ is a pole of the rational map $z \mapsto \mathrm {EF}[\mathbf {U}_0](t,z)$ with $t \in I$ . By possibly enlarging the constant $C>0$ , we obtain the $L^\infty $ -bound with

$$ \begin{align*}\sup_{x \in \mathbb{R}} | \mathrm{EF}[\mathbf{U}_0](t,x) |_F \leq C \quad \text{for } t \in I \,. \end{align*} $$

Since $\mathfrak {H}_1$ has finite dimension, we easily deduce that the degree of the denominator of the rational functions $z \mapsto \mathrm {EF}[\mathbf {U}_0](t,z)$ can be uniformly bounded for $t \in I$ . Hence, by applying Lemma 6.2 below, we deduce

$$ \begin{align*}\sup_{t \in I} \| \mathrm{EF}[\mathbf{U}_0](t) \|_{H^2} \leq C(I, \mathbf{U}_0) \end{align*} $$

with some finite constant $C(I, \mathbf {U}_0)$ .

Since $\Pi _+ \mathbf {V}(t) = \mathrm {EF}[\mathbf {U}_0](t)$ for $t \in [0, T_{\max })$ and by taking a compact interval $I \subset [0,\infty )$ with $[0,T_{\max }) \subset I$ , we conclude that (6.3) holds true. This completes the proof that the maximal (forward) time of existence must be $T_{\max } = +\infty $ .

Finally, by the time reversal symmetry of (HWM $_d$ ) with

$$ \begin{align*}\mathbf{U}(t,x) \mapsto -\mathbf{U}(-t,-x) \, , \end{align*} $$

which maps solutions to solutions (and evidently preserves rationality in x), we deduce that solutions of (HWM $_d$ ) with rational initial data also uniquely extend to all negative times $t \in (-\infty , 0]$ .

This completes the proof of Theorem 1.5.

Proof of Theorem 1.2

This is a direct consequence of Theorem 1.5. Indeed, let $\mathbf {u}_0 \in \mathcal {R}at(\mathbb {R}; \mathbb {S}^2)$ be given and define $\mathbf {U}_0 = \mathbf {u}_0 \cdot \boldsymbol {\sigma } \in \mathcal {R}at(\mathbb {R}; \mathsf {Gr}_1(\mathbb {C}^2))$ . By Theorem 1.5, there exists a unique global solution $\mathbf {U} = \mathbf {U}(t,x)$ of (HWM $_2$ ) with initial datum $\mathbf {U}(0) = \mathbf {u}_0$ . Hence

$$ \begin{align*}\mathbf{u}(t,x) = \frac{1}{2} \mathrm{Tr}( \mathbf{U}(t,x) \boldsymbol{\sigma} )=\frac{1}{2} ( \mathrm{Tr}(\mathbf{U}(t,x) \sigma_1), \mathrm{Tr}(\mathbf{U}(t,x) \sigma_2), \mathrm{Tr}(\mathbf{U}(t,x) \sigma_3)) \end{align*} $$

is the claimed unique global-in-time solution of (HWM) with initial datum $\mathbf {u}(0)=\mathbf {u}_0$ .

We close this section with the following auxiliary result used above.

Lemma 6.2. Let $\mathcal R \subset \mathbb {C}(X)$ be a subset of rational functions. We assume that there exists $C>0$ such that the following properties hold.

1. 1. If $\alpha $ is a pole of some $R\in \mathcal R$ , then

   $$ \begin{align*}|\alpha | +\frac{1}{|\mathrm{Im}(\alpha )|} \leq C\ .\end{align*} $$

2. 2. For every $R\in \mathcal R$ , $R(x)\to 0$ as $x\to \infty $ and

   $$ \begin{align*}\| R \|_{L^\infty (\mathbb{R} )}\leq C\ .\end{align*} $$

3. 3. There exists an integer N such that the degree of the denominator of every $R\in \mathcal R$ is at most N.

Then, for every integer $k\ge 0$ , it holds that

$$ \begin{align*}\sup_{R\in \mathcal R}\| R\|_{H^k(\mathbb{R} )}<\infty \ .\end{align*} $$

Proof. Given $R\in \mathcal R$ , write

$$ \begin{align*}R(x)=\frac{P(x)}{Q(x)}\ ,\ Q(x)=\prod_{j=1}^D (x-\alpha _j)\ ,\ P\in \mathbb{C}[X]\ ,\ \mathrm{deg}(P)<D\leq N\ .\end{align*} $$

Because of properties (1) and (2),

$$ \begin{align*}\max_{0\leq x\leq 1} |P(x)|\leq C\max_{0\leq x\leq 1}\left | \prod_{j=1}^D (x-\alpha _j)\ \right |\leq C(1+C)^N\ .\end{align*} $$

Consequently, all the coefficients $a_j$ of P satisfy

$$ \begin{align*}\sup_{j<D}|a_j|\leq B(N,C)\ ,\end{align*} $$

for some constant $B(N,C)$ depending only on N and C. Similarly, from property (1), all the coefficients of Q are uniformly bounded by a constant depending only on C and N. Moreover, from property (1), for every $x\in \mathbb {R}$ ,

$$ \begin{align*}|Q(x)|\geq \left ( ||x|-C|^2+C^{-2} \right )^{D/2}\ .\end{align*} $$

Notice that the k-th derivative $R^{(k)}$ is a sum of a finite number – depending only on k – of terms of the form

$$ \begin{align*}\frac{ P^{(m)}Q^{(m_1)}\dots Q^{(m_r)} }{Q^{r+1}}\end{align*} $$

where $ 0\leq r\leq k$ and $m+m_1+\dots m_r =k$ . Notice that the degree of the numerator is at most $(r+1)D-k-1$ , and that its coefficients are all bounded by a constant depending only on k, N and C. Consequently,

$$ \begin{align*}\| R^{(k)}\|_{L^2}^2\leq A(k,N,C) \max _{r\leq k}\max _{\ell \leq (r+1)D-k-1}\int_{\mathbb{R}} \frac{x^{2\ell}}{( ||x|-C|^2+C^{-2} )^{D(r+1)}}\, dx\end{align*} $$

with some constant $A(k,N,C)>0$ . This completes the proof.

Preliminaries

Let $d \geq 2$ and $0 \leq k \leq d$ be given integers. In what follows, we suppose that $\mathbf {U}_0 \in \mathcal {R}at(\mathbb {R}; \mathsf {Gr}_k(\mathbb {C}^d))$ holds, that is, the map $\mathbf {U}_0 : \mathbb {R} \to M_d(\mathbb {C})$ is a rational matrix-valued function satisfying the pointwise constraints

In the trivial case of constant initial data $\mathbf {U}_0(x) \equiv \mathbf {U}_\infty $ , we directly obtain Theorem 1.7 with $N=0$ . Hence for the rest of the proof, we will assume that $\mathbf {U}_0$ is nonconstant.

For the following discussion, we need to clearly distinguish between the Toeplitz operator $T_{\mathbf {U}_0}$ acting on the Hardy space $L^2_+(\mathbb {R}; \mathcal {V})$ with $\mathcal {V}=\mathbb {C}^d$ or $\mathcal {V}=M_d(\mathbb {C})$ , respectively.

From Lemma 4.1, we recall the general formula

(7.1) $$ \begin{align} T_{\mathbf{U}_0}^2 = \mathrm{Id} - K_{\mathbf{U}_0} \quad \text{on } L^2_+(\mathbb{R}; \mathcal{V}) \, , \end{align} $$

with the trace-class operator $K_{\mathbf {U}_0} = H_{\mathbf {U}_0}^* H_{\mathbf {U}_0} : L^2_+(\mathbb {R}; \mathcal {V}) \to L^2_+(\mathbb {R}; \mathcal {V})$ . Since $\mathbf {U}_0$ is rational, the operator $K_{\mathbf {U}_0}$ is finite-rank by Lemma 4.3 and we have the finite-dimensional invariant subspace for $T_{\mathbf {U}_0}$ given by

(7.2) $$ \begin{align} \mathfrak{H}_1(\mathcal{V}) := \mathrm{ran} ( K_{\mathbf{U}_0} : L^2_+(\mathbb{R}; \mathcal{V}) \to L^2_+(\mathbb{R}; \mathcal{V}) ) \, , \end{align} $$

where we use the notation $\mathfrak {H}_1(\mathcal {V})$ instead of $\mathfrak {H}_1$ to keep track of whether we choose $\mathcal {V} = \mathbb {C}^d$ or $\mathcal {V} = M_d(\mathbb {C})$ . We introduce the following short-hand notations

(7.3) $$ \begin{align} \mathsf{T}:= T_{\mathbf{U}_0} |_{\mathfrak{H}_1(M_d(\mathbb{C}))} \quad \text{and} \quad \widetilde{\mathsf{T}} := T_{\mathbf{U}_0} |_{\mathfrak{H}_1(\mathbb{C}^d)} \,. \end{align} $$

Note that $\mathsf {T} = \mathsf {T}^*$ and $\widetilde {\mathsf {T}} = \widetilde {\mathsf {T}}^*$ are self-adjoint endomorphisms on the finite-dimensional spaces $\mathfrak {H}_1(M_d(\mathbb {C}))$ and $\mathfrak {H}_1(\mathbb {C}^d)$ , respectively. From Proposition 4.3 we recall that the generator $X^*$ of the adjoint Lax–Beurling semigroup also acts invariantly on the finite-dimensional subspace $\mathfrak {H}_1(\mathcal {V})$ . Likewise, we use the following notation

(7.4) $$ \begin{align} \mathsf{G} := X^* |_{\mathfrak{H}_1(M_d(\mathbb{C}))} \quad \text{and} \quad \widetilde{\mathsf{G}} := X^* |_{\mathfrak{H}_1(\mathbb{C}^d)} \end{align} $$

for the generator $X^*$ of adjoint Lax–Beurling semigroup restricted to the invariant subspaces $\mathfrak {H}_1(M_d(\mathbb {C}))$ and $\mathfrak {H}_1(\mathbb {C}^d)$ , respectively.

Let us now assume $\widetilde {\mathsf {T}}$ has simple spectrum, that is, we have

(7.5) $$ \begin{align} \sigma(\widetilde{\mathsf{T}}) = \{ v_1, \ldots, v_N \} \quad \text{with} \quad N = \dim \mathfrak{H}_1(\mathbb{C}^d) \,. \end{align} $$

Note that $v_n \in (-1,1)$ for $n = 1, \ldots , N$ . Let $\varphi _n \in \mathfrak {H}_1(\mathbb {C}^d) \subset L^2_+(\mathbb {R}; \mathbb {C}^d)$ be a choice of the corresponding normalized eigenfunctions of $\widetilde {\mathsf {T}}$ such that

$$ \begin{align*}\widetilde{\mathsf{T}} \varphi_n = v_n \varphi_n \quad \text{with} \quad \| \varphi_n \|_{L^2} = 1 \end{align*} $$

for $n=1, \ldots , N$ . Clearly, the family $( \varphi _n )_{1 \leq n \leq N}$ forms an orthonormal basis for $\mathfrak {H}_1(\mathbb {C}^d)$ .

We can easily construct an orthonormal basis of eigenfunction for $\mathsf {T}$ acting on the matrix-valued finite-dimensional Hilbert space $\mathfrak {H}_1(M_d(\mathbb {C}))$ as follows. For $1 \leq n \leq N$ and $1 \leq j \leq d$ , we define the matrix-valued functions $\Phi _{n,j} \in L^2_+(\mathbb {R}; M_d(\mathbb {C}))$ by setting

(7.6) $$ \begin{align} \Phi_{n,j}:= \left (0, \ldots, \underbrace{\varphi_n}_{j\text{-th column}}, \ldots, 0 \right ) \,. \end{align} $$

We readily check that

(7.7) $$ \begin{align} \mathsf{T} \Phi_{n,j} = v_n \Phi_{n,j} \quad \text{for } n=1, \ldots, N \text{ and } j=1, \ldots, d \,. \end{align} $$

Thus the eigenvalues $v_n$ for $\mathsf {T}$ are d-fold degenerate in a trivial manner by changing the columns in the matrix-valued functions $\Phi _{n,j}$ .

We have the following fact, whose elementary proof we omit.

Perturbation analysis as $|t| \to \infty $

From Theorem 1.5 we know that the corresponding solution of (HWM $_d$ ) with rational initial datum $\mathbf {U}_0$ is global in time and satisfies

(7.8) $$ \begin{align} \mathbf{U}(t,x) = \mathbf{U}_\infty + \Pi \mathbf{V}(t,x) + (\Pi \mathbf{V}(t,x))^* \, , \end{align} $$

where here and in the following we write $\Pi \equiv \Pi _+$ for the Cauchy–Szegő projection for notational simplicity. By the following explicit flow formula from Lemma 5.2, we have

(7.9) $$ \begin{align} \Pi \mathbf{V}(t,x) = \frac{1}{2 \pi \mathrm{i}} I_+ \left [ (\mathsf{G}+ t \mathsf{T} - x \mathrm{Id})^{-1} \Pi \mathbf{V}_0 \right ] \quad \text{for } (t,x) \in \mathbb{R} \times \mathbb{R} \, , \end{align} $$

using our definitions of $\mathsf {G}$ and $\mathsf {T}$ acting on the finite-dimensional subspace $\mathfrak {H}_1(M_d(\mathbb {C}))$ . Note that we can take $x \in \mathbb {R}$ here, since we have already shown that the rational function $\Pi \mathbf {V}(t,z)$ for $z \in \mathbb {C}_+$ has no poles on the real axis for all $t \in \mathbb {R}$ . Recall also that $\Pi \mathbf {V}_0 \in \mathfrak {H}_1(M_d(\mathbb {C}))$ holds thanks to Proposition 4.2 above.

In order to study the large time limit $t \to \pm \infty $ , it will be convenient to define

(7.10) $$ \begin{align} \mathsf{M}(\varepsilon) := \varepsilon \mathsf{G} + \mathsf{T} \quad \text{with} \quad \varepsilon := \frac{1}{t} \end{align} $$

for $t \neq 0$ . In terms of these definition, we can write the explicit flow formula as

(7.11) $$ \begin{align} \Pi V(\varepsilon^{-1}, x) = \frac{\varepsilon}{2 \pi \mathrm{i}} I_+ [ (\mathsf{M}(\varepsilon)- \varepsilon x \mathrm{Id})^{-1} \Pi V_0 ] \,. \end{align} $$

Inspired by the analysis in [Reference Gérard and Lenzmann15] for the study of N-solitons for the Calogero–Moser derivative NLS, we carry out a perturbation analysis of the non-self-adjoint endomorphisms $\mathsf {M}(\varepsilon ) : \mathfrak {H}_1(M_d(\mathbb {C})) \to \mathfrak {H}_1(M_d(\mathbb {C}))$ in the limit $\varepsilon \to 0$ . We have the following facts, where we recall that we always suppose that the nondegeneracy assumption (7.5) for $\widetilde {\mathsf {T}} : \mathfrak {H}_1(\mathbb {C}^d) \to \mathfrak {H}_1(\mathbb {C}^d)$ holds true.

Proof. We divide the proof of Lemma 7.1 into the following steps.

Step 1. Let $\varepsilon _0> 0$ be a constant chosen later. For $|\varepsilon | \leq \varepsilon _0$ , we define the endomorphisms

(7.12) $$ \begin{align} \widetilde{\mathsf{M}}(\varepsilon) := \widetilde{\mathsf{T}} + \varepsilon \widetilde{\mathsf{G}} : \mathfrak{H}_1(\mathbb{C}^d) \to \mathfrak{H}_1(\mathbb{C}^d) \,. \end{align} $$

Note that $\widetilde {\mathsf {M}}(0) = \widetilde {\mathsf {T}} = \widetilde {\mathsf {T}}^*$ is self-adjoint with simple spectrum $\sigma (\widetilde {\mathsf {T}}) = \{ v_1, \ldots , v_N \}$ with a corresponding orthonormal basis of eigenfunctions $(\varphi _n)_{1 \leq n \leq N}$ . By standard analytic perturbation theory, there exist analytic functions $\varepsilon \mapsto v_n(\varepsilon ) \in \mathbb {C}$ and $\varepsilon \mapsto \varphi _n(\varepsilon ) \in \mathfrak {H}_1(\mathbb {C}^d)$ for $1 \leq n \leq N$ such that

(7.13) $$ \begin{align} \widetilde{\mathsf{T}}(\varepsilon) \varphi_n(\varepsilon) = v_n(\varepsilon) \varphi_n(\varepsilon) \end{align} $$

for $|\varepsilon | \leq \varepsilon _0$ , where $\varepsilon _0> 0$ is some sufficiently small constant. We have

(7.14) $$ \begin{align} v_n(\varepsilon) = v_n + \varepsilon w_n + O(\varepsilon^2), \quad \varphi_n(\varepsilon) = \varphi_n + O(\varepsilon) \, , \end{align} $$

(7.15) $$ \begin{align} w_n = \langle \widetilde{\mathsf{G}} \varphi_n, \varphi_n \rangle \,. \end{align} $$

Since $(\varphi _n)_{1 \leq n \leq N}$ forms an orthonormal basis for $\mathfrak {H}_1(\mathbb {C}^d)$ and by continuity with respect to $\varepsilon $ , we readily see that the perturbed eigenvectors $(\varphi _n(\varepsilon ))_{1 \leq n \leq N}$ also form a (not necessarily orthonormal) basis of $\mathfrak {H}_1(\mathbb {C}^d)$ , provided that $\varepsilon _0> 0$ is sufficiently small. By defining

$$ \begin{align*}\Phi_{n,j}(\varepsilon) := \left (0, \ldots, \underbrace{\varphi_n(\varepsilon)}_{j\text{-th column}}, \ldots, 0 \right ) \, , \end{align*} $$

we easily verify that (i) holds true.

Step 2. It remains to prove item (ii). Thus we claim, for any $1 \leq n \leq N$ ,

(7.16) $$ \begin{align} \mathrm{Im} \, w_n = \mathrm{Im} \, \langle \widetilde{\mathsf{G}} \varphi_n, \varphi_n \rangle < 0 \,. \end{align} $$

Indeed, let $1 \leq n \leq N$ be given. From the general identity (2.3), we recall that

(7.17) $$ \begin{align} \mathrm{Im} \, \langle \widetilde{\mathsf{G}} \varphi_n, \varphi_n \rangle = - \frac{1}{4 \pi} |I_+(\varphi_n)|^2 \leq 0 \, , \end{align} $$

with $I_+(f) = \lim _{\xi \to 0^+} \widehat {f}(\xi )$ and $f \in \mathrm {dom}(X^*)$ . [Note that $\varphi _n \in \mathrm {dom}(X^*)$ , since $\varphi _n$ is a rational function.] To prove (7.16), we argue by contradiction as follows. Let us assume that

(7.18) $$ \begin{align} I_+(\varphi_n) = 0 \,. \end{align} $$

By the commutator formula in Lemma 3.2, we deduce

(7.19) $$ \begin{align} [\widetilde{\mathsf{G}}, \widetilde{\mathsf{T}}] \varphi_n = \frac{\mathrm{i}}{2 \pi} \Pi \mathbf{V}_0.I_+(\varphi_n) = 0 \,. \end{align} $$

Thus from $\widetilde {\mathsf {T}} \varphi _n = v_n \varphi _n$ we see that $\widetilde {\mathsf {T}} \widetilde {\mathsf {G}} \varphi _n = v_n \widetilde {\mathsf {G}} \varphi _n$ . But since $\widetilde {\mathsf {T}}$ has simple spectrum by assumption, we conclude $\widetilde {\mathsf {G}} \varphi _n = \alpha \varphi _n$ for some constant $\alpha \in \mathbb {C}$ . By taking the Fourier transform, this yields

(7.20) $$ \begin{align} \mathrm{i} \frac{d}{d \xi} \widehat{\varphi}_n(\xi) = \alpha \widehat{\varphi}_n(\xi) \quad \text{for } \xi \geq 0 \,. \end{align} $$

Thus we find $\widehat {\varphi }(\xi ) = A \mathrm {e}^{-\mathrm {i} \alpha \xi }$ for $\xi \geq 0$ with some constant $A \neq 0$ (since $\varphi _n \not \equiv 0$ ). Moreover, we infer that $\mathrm {Im} \, \alpha < 0$ since $\widehat {\varphi }_n \in L^2(\mathbb {R}_+)$ . But this implies that

$$ \begin{align*}I_+(\varphi_n) = \lim_{\xi \to 0^+} \widehat{\varphi}_n(\xi) = A \neq 0 \, , \end{align*} $$

contradicting our assumption that $I_+(\varphi _n)=0$ holds. This shows (7.16) and completes the proof of Lemma 7.1.

Proof of Theorem 1.7

We are now ready to give the proof of Theorem 1.7. Adapting the notation from above, we proceed as follows.

Asymptotic behavior as $t \to \pm \infty $

Recall that $\varepsilon = t^{-1}$ for $t \neq 0$ . In what follows, we shall always assume that $|\varepsilon | \leq \varepsilon _0$ with the constant $\varepsilon _0> 0$ from Lemma 7.1 above, which amounts to considering times t with $|t| \geq T_0$ where $T_0 = \varepsilon ^{-1}$ .

We expand $\Pi \mathbf {V}_0 \in \mathfrak {H}_1(M_d(\mathbb {C}))$ in terms of the basis $(\Phi _{n,j}(\varepsilon ))_{1 \leq n \leq N, 1 \leq j \leq d}$ from Lemma 7.1, that is, we write

(7.21) $$ \begin{align} \Pi \mathbf{V}_0 = \sum_{n=1}^N \sum_{j=1}^d \alpha_{n,j}(\varepsilon) \Phi_{n,j}(\varepsilon) \end{align} $$

with some coefficients $\alpha _{n,j}(\varepsilon ) \in \mathbb {C}$ . From Lemma 7.1 and the fact that $\varepsilon x \in \mathbb {R}$ does not belong to the spectrum $\sigma (\mathsf {M}(\varepsilon )) = \{ v_1(\varepsilon ), \ldots , v_N(\varepsilon ) \} \subset \mathbb {C}_-$ , we conclude that

(7.22) $$ \begin{align} (\mathsf{M}(\varepsilon) - \varepsilon x \mathrm{Id})^{-1} \Pi \mathbf{V}_0 = \sum_{n=1}^N \sum_{j=1}^d \frac{\alpha_{n,j}(\varepsilon)}{v_n(\varepsilon) - \varepsilon x} \Phi_{n,j}(\varepsilon) \quad \text{for } |\varepsilon| \leq \varepsilon_0 \text{ and } x \in \mathbb{R} \,. \end{align} $$

In view of the explicit formula (7.11) together with $\varepsilon = t^{-1}$ and the properties stated in Lemma 7.1, we obtain that

(7.23) $$ \begin{align} \Pi \mathbf{V}(t,x) = \frac{\varepsilon}{2 \pi \mathrm{i}} I_+ \left [ \sum_{n=1}^N \sum_{j=1}^d \frac{\alpha_{n,j}(\varepsilon)}{v_n(\varepsilon) - \varepsilon x} \Phi_{n,j}(\varepsilon) \right ] = \sum_{n=1}^N \frac{A_n(t)}{x-z_n(t)} \,. \end{align} $$

Here we set

(7.24) $$ \begin{align} z_n(t) := t v_n(t^{-1}) = t v_n + w_n + O(t^{-1}) \in \mathbb{C}_- \quad \text{for } |t| \geq T_0 \, , \end{align} $$

and $A_n(t) \in M_d(\mathbb {C})$ are the matrix-valued functions defined as

(7.25) $$ \begin{align} A_n(t) := - \frac{1}{2 \pi \mathrm{i}} \sum_{j=1}^d \alpha_{n,j}(t^{-1}) I_+[\Phi_{n,j}(t^{-1})] \quad \text{for } |t| \geq T_0 \,. \end{align} $$

Note that, by choosing $T_0> 0$ possibly larger, we can henceforth ensure that

(7.26) $$ \begin{align} |z_n(t) - z_m(t)| \geq \frac{|t|}{2} \cdot \min_{n \neq m} |v_n - v_m|> 0 \quad \text{for } |t| \geq T_0 \text{ and } n \neq m, \end{align} $$

implying that the poles $z_1(t), \ldots , z_N(t) \in \mathbb {C}_-$ are pairwise distinct whenever $|t| \geq T_0$ .

Next we show, after discarding possibly trivial zero terms, that all the matrices $A_j(t) \in M_d(\mathbb {C})$ are nonzero and nilpotent of degree 2. Moreover, their limits as $t \to \pm \infty $ both exist, coincide, and are nonzero as well.

Proposition 7.2. There exists an integer $1 \leq M \leq N$ such that, after possibly relabelling $ \{ A_n(t), z_n(t) \}_{n=1}^N$ , it holds that

$$ \begin{align*}\Pi_+ \mathbf{V}(t,x) = \sum_{n=1}^M \frac{A_n(t)}{x-z_n(t)} \quad \text{for } |t| \geq T_0 \text{ and } x \in \mathbb{R} \,. \end{align*} $$

Here the matrices $A_n(t) \in M_d(\mathbb {C})$ satisfy $A_n(t) \neq 0$ and $A_n(t)^2 = 0$ for $|t| \geq T_0$ .

In addition, it holds

$$ \begin{align*}A_n(t) = A_n + O(t^{-1}) \end{align*} $$

with some nonzero limits $A_n \neq 0$ satisfiying $A_n^2 = 0$ for $1 \leq n \leq M$ .

Remark. In the special case of (HWM) with target $\mathbb {S}^2$ , corresponding to the target $\mathsf {Gr}_1(\mathbb {C}^2)$ in the matrix-valued case, we will see below that actually $M=N$ must always hold. This observation is based on the simple algebraic fact that nonzero matrices $A \in M_2(\mathbb {C})$ with $A^2=0$ must have $\mathrm {rank}(A) = 1$ . See below for more details.

Proof. Step 1. We first show that $A_n(t)^2 = 0$ holds for $1 \leq n \leq N$ and $|t| \geq T_0$ . Indeed, we know that

(7.27) $$ \begin{align} \mathbf{U}(t,x) = \mathbf{U}_\infty + \sum_{n=1}^N \frac{A_n(t)}{x-z_n(t)} + \sum_{n=1}^N \frac{A_n(t)^*}{x-\overline{z}_n(t)} \quad \text{for } |t| \geq T_0 \end{align} $$

with the pairwise distinct poles $z_1(t), \ldots , z_n(t) \in \mathbb {C}_-$ and some constant matrix $\mathbf {U}_\infty \in \mathsf {Gr}_k(\mathbb {C}^d)$ . From the algebraic constraint and by equating the terms proportional to $(x-z_n(t))^{-2}$ to zero, we conclude that

$$ \begin{align*}A_n(t)^2 = 0 \end{align*} $$

for $|t| \geq T_0$ and $1 \leq n \leq N$ .

Furthermore, we readily see that we have existence and equality of the limits

$$ \begin{align*}\lim_{t \to -\infty} A_n(t) = \lim_{t \to +\infty} A_n(t) =: A_n \in M_d(\mathbb{C}) \,. \end{align*} $$

This directly follows from the properties in Lemma 7.1 which yields that

$$ \begin{align*}\lim_{|t| \to \infty} A_n(t) = -\frac{1}{2 \pi \mathrm{i}} \lim_{\varepsilon \to 0} \sum_{j=1}^d \alpha_{n,j}(t^{-1}) I_+[\Phi_{n,j}(t^{-1})] = -\frac{1}{2 \pi \mathrm{i}} \sum_{j=1}^d \alpha_{n,j} I_+[\Phi_{n,j}] = A_n \end{align*} $$

with the coefficients $\alpha _{n,j} = \langle \Pi \mathbf {V}_0, \Phi _{n,j} \rangle $ . Moreover, since $\Phi _{n,j}(t^{-1}) = \Phi _{n,j} + O(t^{-1})$ , we readily deduce that

$$ \begin{align*}A_n(t) = A_n + O(t^{-1}) \,. \end{align*} $$

Moreover, from $A_n(t)^2 = 0$ for $|t| \geq T_0$ , we readily deduce that the limits satisfy $A_n^2=0$ as well.

Step 2. By plugging (7.27) into (HWM $_d$ ), we obtain the following differential equations for the matrix-valued functions $A_n(t)$ :

(7.28) $$ \begin{align} \dot{A}_n(t) = \mathrm{i} \sum_{m \neq n}^N \frac{[A_n(t), A_m(t)]}{(z_n(t)-z_m(t))^2} \quad \text{for } |t| \geq T_0 \text{ and } 1 \leq n \leq N \, , \end{align} $$

where $[X,Y]$ denotes the commutator of matrices in $M_d(\mathbb {C})$ . For details of the calculation that derives (7.28), we refer to the proof of [Reference Berntson, Klabbers and Langmann3][Theorem 2.1]; the generalization to (HWM $_d$ ) is straightforward. We also note that the expression on the right-hand side in (7.28) is nonsingular for $|t| \geq T_0$ thanks to (7.26).

We now claim that

(7.29) $$ \begin{align} A_n(T_0) \neq 0 \quad \Rightarrow \quad A_n(t) \neq 0 \text{ for } t \geq T_0 \text{ and } \displaystyle \lim_{t \to +\infty} A_n(t) \neq 0 \,. \end{align} $$

Indeed, let $\| A \| = (\mathrm {Tr}(A A^*))^{1/2}$ denote the Frobenius norm of a matrix $A \in M_d(\mathbb {C})$ . Since $\| A_m(t) \| \leq C$ for $t \geq T_0$ and $1 \leq m \leq N$ with some constant $C>0$ (by existence of limits shown in Step 1) and from (7.26), we obtain from (7.28) the estimate

(7.30) $$ \begin{align} \| \frac{d}{dt} A_n(t) \| \lesssim \frac{1}{t^2} \| A_n(t) \| \quad \text{for } t \geq T_0 \,. \end{align} $$

Suppose now that $A_n(T_0) \neq 0$ and let $T \in (T_0,+\infty ]$ . Then by integrating the estimate above, we conclude that

$$ \begin{align*}\int_{T_0}^T \frac{d}{dt} \log \|A_n(t)\| \, dt = \log(\| A_n(T) \| ) - \log(\| A_n(T_0) \|) \lesssim \int_{T_0}^T \frac{dt}{t^2} \lesssim \frac{1}{T_0} < +\infty\end{align*} $$

which rules out $A_n(T) = 0$ for $T \in (T_0, +\infty ]$ . This proves the implication (7.29).

Step 3. Define the integer $0 \leq K \leq N$ by setting

$$ \begin{align*}K := \# \{ 1 \leq n \leq N : A_n(T_0) = 0\} \end{align*} $$

and we let $M := N-K$ . Now if $M=0$ , then $\mathbf {U}(t,x) = \mathbf {U}_0 = \mathbf {U}_\infty $ is a constant solution to (HWM $_d$ ). But this implies that $K_{\mathbf {U}_0} \equiv 0$ and hence $\mathfrak {H}_1(\mathbb {C}^d) = \{ 0 \}$ is trivial, which contradicts our assumption that $N = \dim \mathfrak {H}_1(\mathbb {C}^d) \geq 1$ . Thus we see that $M \geq 1$ holds.

Thus, after relabelling $\{ A_n(T_0), z_n(T_0) \}_{n=1}^N$ if necessary, we see that

$$ \begin{align*}\Pi \mathbf{V}(T_0, x) = \sum_{n=1}^M \frac{A_n(T_0)}{x-z_n(T_0)} \end{align*} $$

with $A_n(T_0) \neq 0$ for $1 \leq n \leq M$ . By (7.29), we deduce that $A_n(t) \neq 0$ for all $t \geq T_0$ and $1 \leq n \leq M$ and $\lim _{t \to +\infty } A_n(t) = A_n \neq 0$ for $1 \leq n \leq M$ . This proves statement of Proposition 7.2 for positive times $t \geq T_0$ .

Finally, since $\lim _{t \to -\infty } A_n(t) = \lim _{t \to +\infty } A_n(t)$ for all $1 \leq n \leq N$ by Step 1, we complete the proof of Proposition 7.2 for negative times $t \leq -T_0$ .