# KdV hierarchy via Abelian coverings and operator identities

## Abstract

We establish precise spectral criteria for potential functions of reflectionless Schrödinger operators to admit solutions to the Korteweg–de Vries (KdV) hierarchy with 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

and the associated hierarchy of higher-order differential equations

for which we will define precisely below. For the sake of introduction, we simply note that , is the typical KdV equation, and that is generally an order polynomial differential operator. We establish conditions on in terms of the *spectral properties* of the associated Schrödinger operator 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 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 up to a certain order under our conditions. -functions

### 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 on it. The Baker-Akhiezer function contains two factors: the first one is given by an Abelian integral with a certain singularity at 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 although we are restricted by the assumption that , are real. In our normalization and does not contain any isolated points. So, we consider the domain

which would play a role of the “upper sheet” for a compact hyperelliptic surface , 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 be the Green function in the domain with logarithmic singularity at We assume that . is continuous up to the boundary; that is, 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 is related to the Green function by

Letting be the fundamental group of the given domain, analytic continuation of the function along the path picks up a unimodular multiplicative factor

In this case we say is (multiplicative) character automorphic with character .

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 associated to an arbitrary unitary character of the group .

This space possesses a reproducing kernel, which we denote by That is, .

Since the elements of 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 .

Recall that for the given domain there exists a function analytic in the upper half-plane and a Fuchsian group such that induces a one-to-one correspondence between the points and orbits , We will normalize this function by the conditions .

Due to this normalization, is defined up to a positive multiplier. By we denote the group of unitary characters of the group .

The following definition is basically parallel to Definition 1.2.

The spaces and correspond by way of the uniformization .

We denote the reproducing kernel in by 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 is the Widom condition, which is necessary and sufficient for non-triviality of the spaces for all .

- (PW)
We assume that for some (and hence all) we have ,

As soon as Equation 1.8 holds, is called a domain of Widom type, and is respectively called a group of Widom type. For all critical values are real; moreover, there is exactly one critical point in each gap, .

The Widom condition (PW) along with our third assumption allows us to extend the notion of Abelian integrals of the second kind:

- ( -GLC)
By the order gap length condition we mean

Under the assumptions (PW) and ( we can define the following (see Proposition -GLC),4.1 below).

As before, the analytic continuation along the path is related to the original value by

We say that is an additive character on and that the function is (additive) character automorphic.

In the special case we will drop the index. 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 is defined as follows: denote by the commutator subgroup of

The quotient is canonically dual to by Pontryagin duality. Thus, any of the multivalued analytic continuations defined above become functions on the surface In particular, we have the covering map . :

Similarly, we consider generalized Abelian integrals as functions on and denote them by As one would expect, the group . acts on this surface by

where

The Main Theorem claims that the KdV hierarchy equation of order *multiplication operators* for the functions *commute* as actions on a Hardy space

We require such notions as inner and outer functions, and functions of bounded characteristic (of class

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

The action of the group

Note that essentially

The space

Consider the collection of functions