KdV hierarchy via Abelian coverings and operator identities
By B. Eichinger, T. VandenBoom, and P. Yuditskii
Abstract
We establish precise spectral criteria for potential functions $V$ of reflectionless Schrödinger operators $L_V = -\partial _x^2 + V$ to admit solutions to the Korteweg–de Vries (KdV) hierarchy with $V$ as an initial value. More generally, our methods extend the classical study of algebro-geometric solutions for the KdV hierarchy to noncompact Riemann surfaces by defining generalized Abelian integrals and analogues of the Baker-Akhiezer function on infinitely connected domains with a uniformly thick boundary satisfying a fractional moment condition.
1. Introduction
We study the Cauchy problem for the Korteweg–de Vries (KdV) equation
$$\begin{align*} \partial _t V &= \frac{1}{4}\partial _x^3 V - \frac{3}{2}V\partial _x V, \\ V(\cdot , 0) &= V(\cdot ) \end{align*}$$
and the associated hierarchy of higher-order differential equations
for $k \in {\mathbb{N}}$, which we will define precisely below. For the sake of introduction, we simply note that $\operatorname {KdV}_1(V) = \frac{1}{4}\partial _x^3 V - \frac{3}{2}V\partial _x V$ is the typical KdV equation, and that $\operatorname {KdV}_k$ is generally an order $2k+1$ polynomial differential operator. We establish conditions on $V$ in terms of the spectral properties of the associated Schrödinger operator $L_V = -\partial _x^2 + V$ which guarantee global existence and temporal and spatial almost-periodicity of classical solutions.
We present this result first because of its broad appeal. Recent work Reference 1 of Binder, Damanik, Goldstein, and Lukic has, under stronger assumptions, proven something similar in the case $k = 1$, and Kotani has announced a related result under an integer moment condition Reference 16. In our case, Theorem 1.1 is only a facet of our Main Theorem, Theorem 1.14. In fact, the full content of this paper is the development of a spectrally dependent Fourier transform, with respect to which the study of reflectionless Schrödinger operators becomes greatly simplified.
Naturally, our result is based on Lax pair representation in the theory of integrable systems (see e.g. Reference 6Reference 11), the spectral theory of ergodic 1-D Schrödinger operators (see e.g. Reference 25), and the functional model approach to the spectral theory (see e.g. Reference 23). The relationship between the KdV equation and the Schrödinger operator via the Lax pair formalism was noted in the 1960s by Lax Reference 19. At the same time, use of inverse scattering techniques to solve the KdV equation was pioneered by Gardner, Greene, Kruskal, and Miura Reference 8. The 1970s saw significant further development expounding on these ideas to solve the KdV equation for periodic initial data Reference 4Reference 21Reference 24. Shortly thereafter, algebro-geometric extensions of the techniques from the periodic setting were developed to address almost-periodic and ergodic initial conditions having finite-gap spectra; see Reference 11 for a textbook treatment on this approach. Extensions to the infinite-gap setting have been partially developed in important work by Egorova Reference 7, Kotani Reference 17, and more recently in work of Binder, Damanik, Goldstein, and Lukic Reference 1. Some of our methods, including the ideas of generalized Abelian integrals and conformal mappings onto comb domains, were established by Marchenko in Reference 20. Other important methods in integrable systems were developed by Deift and Zhao Reference 2.
The sequel will be structured as follows: the remainder of Section 1 establishes definitions and notation en route to our main theorem, Theorem 1.14. Section 2 recalls some basic preliminaries from spectral theory and establishes the spectral infimum as a normalization point. Section 3 describes in detail the functional models and proves an important identity regarding the reproducing kernels. Finally, Section 4 discusses the generalized Abelian integrals and their relationship to the finite-gap case and proves the asymptotic expansion of the $m$-functions up to a certain order under our conditions.
1.1. Generalized Abelian integrals and Hardy spaces on infinitely connected domains
Algebro-geometric solutions for the KdV hierarchy equations are given by means of the Baker-Akhiezer function; see e.g. Reference 18Reference 29, and books Reference 6Reference 22; see also Reference 11. This function is associated to a hyperelliptic Riemann surface (with the standard compactification)
and a specified point $\infty \in {\mathcal{S}}$ on it. The Baker-Akhiezer function contains two factors: the first one is given by an Abelian integral with a certain singularity at $\infty$, and the second one is represented as a specific ratio of theta-functions closely related to the so-called prime form Reference 22, IIIb, §1.
Our Baker-Akhiezer function is Equation 1.19. It allows for the much more general case when $N=\infty$, although we are restricted by the assumption that $a_j,b_j$ are real. In our normalization $0<a_j<b_j$ and $E:={\mathbb{R}}_+\setminus \bigcup _{j\ge 1}(a_j,b_j)$ does not contain any isolated points. So, we consider the domain
$$\begin{equation*} {\mathcal{S}}_+={\mathbb{C}}\setminus E, \quad \end{equation*}$$
which would play a role of the “upper sheet” for a compact hyperelliptic surface ${\mathcal{S}}$,$N<\infty$. A classical construction of Abelian integrals is given by means of potential theory Reference 14, Part III, Ch. 9. Our first assumption is that our domain is standard in this regard:
(R)
Let $G_{\lambda _0}(\lambda )=G(\lambda ,\lambda _0)$ be the Green function in the domain ${\mathcal{S}}_+$ with logarithmic singularity at $\lambda _0\in {\mathcal{S}}_+$. We assume that $G_{\lambda _0}$ is continuous up to the boundary; that is, ${\mathcal{S}}_+$ is regular in terms of the potential theory Reference 10, Theorem 6.3, p. 95.
An interpretation of an Abelian integral of the third kind in terms of the Green function is well known; specifically, the generalized Abelian integral of the third kind $-\log \Phi (\lambda ,\lambda _0)$ is related to the Green function by
Letting $\pi _1({\mathcal{S}}_+)$ be the fundamental group of the given domain, analytic continuation of the function $\Phi _{\lambda _0}(\lambda )=\Phi (\lambda ,\lambda _0)$ along the path $\gamma \in \pi _1({\mathcal{S}}_+)$ picks up a unimodular multiplicative factor
In this case we say $\Phi _{\lambda _0}(\lambda )$ is (multiplicative) character automorphic with character $\nu _{\lambda _0}$.
The counterpart of the theta-function is hardly possible in our level of generality, but the ratio of two of them can have perfect sense; we suggest treating it as a special function associated to the problem. We interpret the prime form (the second factor in our generalized Baker-Akhiezer function) as the reproducing kernel of a suitable Hilbert space of analytic functions. By “suitable Hilbert spaces” we mean Hardy spaces $H^2_{{\mathcal{S}}_+}(\alpha )$ associated to an arbitrary unitary character $\alpha$ of the group $\pi _1({\mathcal{S}}_+)$.
This space possesses a reproducing kernel, which we denote by $k^\alpha _{{\mathcal{S}}_+}(\lambda ,\lambda _0)=k^\alpha _{{\mathcal{S}}_+,\lambda _0}(\lambda )$. That is,
Since the elements of $H^2_{{\mathcal{S}}_+}(\alpha )$ are multivalued functions, Equation 1.7 creates a state of small uncertainty. To avoid this uncertainty, we provide an alternative definition using the universal covering for ${\mathcal{S}}_+$.
Recall that for the given domain there exists a function ${\boldsymbol{\lambda }}(z)$ analytic in the upper half-plane ${\mathbb{C}}_+$ and a Fuchsian group $\Gamma \simeq \pi _1({\mathcal{S}}_+)$ such that ${\boldsymbol{\lambda }}(z)$ induces a one-to-one correspondence between the points $\lambda \in {\mathcal{S}}_+$ and orbits $\{\gamma (z)\}_{\gamma \in \Gamma }$,$z\in {\mathbb{C}}_+$. We will normalize this function by the conditions
Due to this normalization, ${\boldsymbol{\lambda }}(z)$ is defined up to a positive multiplier. By $\Gamma ^*$ we denote the group of unitary characters of the group $\Gamma$.
The following definition is basically parallel to Definition 1.2.
The spaces $H^2_{{\mathcal{S}}_+}(\alpha )$ and $H^2_\Gamma (\alpha )$ correspond by way of the uniformization ${\boldsymbol{\lambda }}(z)$.
We denote the reproducing kernel in $H_{\Gamma }^2(\alpha )$ by $k_{\Gamma }^\alpha (z,z_0)$ and remark that the correspondence in Proposition 1.4 completely removes the above-mentioned ambiguity in Equation 1.7. For a textbook discussion of character automorphic Hardy spaces, see Reference 12.
Our second assumption on ${\mathcal{S}}_+$ is the Widom condition, which is necessary and sufficient for non-triviality of the spaces $H_{\Gamma }^2(\alpha )$ for all $\alpha \in \Gamma ^*$.
(PW)
We assume that for some (and hence all) $\lambda _0 \in {\mathcal{S}}_+$, we have$$\begin{equation} \sum _{\nabla G_{\lambda _0}(\xi )=0} G(\xi ,\lambda _0)<\infty . \cssId{texmlid6}{\tag{1.8}} \end{equation}$$
As soon as Equation 1.8 holds, ${\mathcal{S}}_+$ is called a domain of Widom type, and $\Gamma$ is respectively called a group of Widom type. For $\lambda _0\in {\mathbb{R}}_-$ all critical values $\xi =\xi (\lambda _0)$ are real; moreover, there is exactly one critical point in each gap, $\xi _j(\lambda _0)\in (a_j,b_j)$.
The Widom condition (PW) along with our third assumption allows us to extend the notion of Abelian integrals of the second kind:
($k$-GLC)
By the order $k$ gap length condition we mean$$\begin{equation} \int _{{\mathbb{R}}_+\setminus E}\mathrm{d}\xi ^{k+\frac{1}{2}} =\sum _{j\ge 1}(b_j^{k+\frac{1}{2}}-a_j^{k+\frac{1}{2}})<\infty . \cssId{texmlid13}{\tag{1.9}} \end{equation}$$
Under the assumptions (PW) and ($k$-GLC), we can define the following (see Proposition 4.1 below).
As before, the analytic continuation $\Theta ^{(k)}(\gamma (\lambda ))$ along the path $\gamma \in \pi _1({\mathcal{S}}_+)$ is related to the original value $\Theta ^{(k)}(\lambda )$ by
We say that $\eta ^{(k)}$ is an additive character on $\pi _1({\mathcal{S}}_+)$ and that the function $\Theta ^{(k)}(\lambda )$ is (additive) character automorphic.
In the special case $k=0$ we will drop the index. $M(\lambda )$ is called the Martin function of the given domain (with respect to infinity) or Phragmén-Lindelöf function (particularly with respect to the representation Equation 1.10; see Reference 15, Theorem, p. 407).
1.2. Functional models on the universal Abelian cover in application to spectral theory of 1-D Schrödinger operators
We can now trace the path towards our Main Theorem. The universal Abelian covering of ${\mathcal{S}}_+$ is defined as follows: denote by $\Gamma '$ the commutator subgroup of $\Gamma$
The quotient $\Gamma /\Gamma '$ is canonically dual to $\Gamma ^*$ by Pontryagin duality. Thus, any of the multivalued analytic continuations defined above become functions on the surface ${\mathcal{R}}:={\mathbb{C}}_+/\Gamma '$. In particular, we have the covering map $\lambda _{{\mathcal{R}}}:{\mathcal{R}}\to {\mathcal{S}}_+$:
Similarly, we consider generalized Abelian integrals as functions on ${\mathcal{R}}$ and denote them by $\theta _{{\mathcal{R}}}^{(k)}(p)=\Theta ^{(k)}(\lambda _{{\mathcal{R}}}(p))$. As one would expect, the group $\Gamma /\Gamma '$ acts on this surface by
where $\underline{\gamma }\in \Gamma /\Gamma '$ denotes the equivalence class in $\Gamma /\Gamma '$ of an element $\gamma \in \Gamma$. According to this notation Equation 1.11 has the form
The Main Theorem claims that the KdV hierarchy equation of order $k$ is simply the relation that the two multiplication operators for the functions $\lambda _{{\mathcal{R}}}(p)$ and $\theta ^{(k)}_{{\mathcal{R}}}(p)$commute as actions on a Hardy space $H^2_{{\mathcal{R}}}$ associated to the Abelian cover ${\mathcal{R}}$; we define this space below.
We require such notions as inner and outer functions, and functions of bounded characteristic (of class ${\mathcal{N}}$) and of its Smirnov subclass (or Nevanlinna class ${\mathcal{N}}_+$) on the Riemann surfaces. These objects are well known in the theory of Hardy spaces in the disc ${\mathbb{D}}$ or half-plane ${\mathbb{C}}_+$Reference 9; particularly the class ${\mathcal{N}}_+$ is defined in Reference 9, Ch. II, Sect. 5. We say that the function is outer (inner) on ${\mathcal{S}}_+$ or ${\mathcal{R}}$ if its lift to the universal cover ${\mathbb{C}}_+$ is outer (inner). We say that $F(p)$,$p\in {\mathcal{R}}$, is of bounded characteristic if it can be represented as a ratio of two bounded analytic functions. It is of Smirnov class if in addition the denominator is an outer function.
The integral density of states is a fundamental measure in the spectral theory of ergodic operators. In our case it coincides with a renormalization of $\mathrm{d}\Theta$ restricted on $E = {\partial }{\mathcal{S}}_+$. Above we paid tribute to its importance in our definition of $H_{{\mathcal{R}}}^2$ as a subspace of $L^2$-space with respect to $\mathrm{d}\theta _{{\mathcal{R}}}$ on the boundary ${\partial }{\mathcal{R}}$.
The action of the group $\Gamma /\Gamma '$ on ${\partial }{\mathcal{R}}$ is dissipative. This means that there exists a fundamental measurable set ${\mathbb{E}}\subset {\partial }{\mathcal{R}}$ such that $\underline{\gamma }{\mathbb{E}}\cap {\mathbb{E}}=\emptyset$ for $\underline{\gamma }\not =1_{\Gamma /\Gamma '}$, and, for an arbitrary $L^1$-function$F(p)$,
Note that essentially ${\mathbb{E}}={\partial }{\mathcal{S}}_+$. We define the Hardy spaces $H^2(\alpha )$ of character automorphic functions (with respect to the action of the group $\Gamma /\Gamma '$ on ${\mathcal{R}}$) in the following way.
The space $H^2(\alpha )$ is a small modification of the spaces $H_{{\mathcal{S}}_+}^2(\alpha ) \cong H^2_\Gamma (\alpha )$ defined above; for a precise relationship between these spaces, see Lemma 3.3. The advantage of considering these spaces in this new way is that we can collect all character automorphic Hardy spaces $H^2(\alpha )$ within $H^2_{\mathcal{R}}$ in a sense we now describe.
Consider the collection of functions $f(p,\alpha )$ such that $f(p,\alpha )\in H^2(\alpha )$ for a.e. $\alpha \in \Gamma ^*$ with respect to the Haar measure $\mathrm{d}\alpha$ on $\Gamma ^*$ and