Unique Solvability of a Coupling Problem for Entire Functions
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Abstract
We establish the unique solvability of a coupling problem for entire functions that arises in inverse spectral theory for singular second-order ordinary differential equations/two-dimensional first-order systems and is also of relevance for the integration of certain nonlinear wave equations.
Keywords
Coupling problem for entire functions Unique solvability Inverse spectral theoryMathematics Subject Classification
Primary 30D20 34A55 Secondary 34B05 37K151 Results
Coupling problem
- (C)
- Coupling condition:^{1}$$\begin{aligned} \Phi _-(\lambda ) = \eta (\lambda ) \Phi _+(\lambda ), \quad \lambda \in \sigma \end{aligned}$$
- (G)
- Growth and positivity condition:$$\begin{aligned} {\mathrm {Im}}\biggl ( \frac{z \Phi _-(z) \Phi _+(z)}{W(z)}\biggr ) \ge 0, \quad {\mathrm {Im}}(z)>0 \end{aligned}$$
- (N)
- Normalization condition:$$\begin{aligned} \Phi _-(0) = \Phi _+(0) = 1 \end{aligned}$$
Definition
The main purpose of the present article is to prove that this simple condition is sufficient to guarantee unique solvability of the corresponding coupling problem.
Theorem
(Existence and Uniqueness) If the coupling constants \(\eta \in \hat{{\mathbb {R}}}^\sigma \) are admissible, then the coupling problem with data \(\eta \) has a unique solution.
Apart from this result, we will also establish the fact that the solution of the coupling problem depends in a continuous way on the given data.
Proposition
(Stability) Let \(\eta _k\in \hat{{\mathbb {R}}}^\sigma \) be a sequence of admissible coupling constants that converge to some coupling constants \(\eta \) (in the product topology). Then the solutions of the coupling problems with data \(\eta _k\) converge locally uniformly to the solution of the coupling problem with (admissible) data \(\eta \).
In the simple case when the set \(\sigma \) consists of only one point, we are able to write down solutions explicitly in terms of the single coupling constant.
Example
As we will see in the course of the proofs, it is still possible to construct solutions of coupling problems when the set \(\sigma \) is only assumed to be finite although the situation is considerably more intricate. These explicit solutions can then be utilized to approximate solutions of coupling problems on infinite sets \(\sigma \).
Even though in the example above the coupling problem is solvable if and only if the coupling constants are admissible, this is not the case in general. Indeed, it is not too difficult to construct counterexamples for this as soon as the set \(\sigma \) contains more than one point. The following observation sheds some light on what happens in the situation when the coupling constants are not necessarily admissible.
Remark
Before we proceed to the proofs of our results, let us point out two applications that constitute our main motivation for considering this coupling problem for entire functions. First and foremost, the coupling problem is essentially equivalent to an inverse spectral problem for second-order ordinary differential equations or two-dimensional first-order systems with trace class resolvents. This circumstance indicates that it is not likely for a simple elementary proof of our theorem to exist, as the uniqueness part allows one to effortlessly deduce (generalizations of) results in [3, 7, 13, 14, 19], which had to be proven in a more cumbersome way before. On the other hand, the coupling problem is also of relevance for certain completely integrable nonlinear wave equations (with the Camassa–Holm equation [4, 9] and the Hunter–Saxton equation [21] being the prime examples) when the underlying isospectral problem has purely discrete spectrum. For these kinds of equations, the coupling problem takes the same role as Riemann–Hilbert problems do in the case when the associated spectrum has a continuous component; see [1, 8, 11]. In particular, the stability result for the coupling problem enables us to derive long-time asymptotics for solutions of such nonlinear wave equations [18].
1.1 Inverse Spectral Theory
1.2 Nonlinear Wave Equations
2 Proofs
Lemma A
- (i)The de Branges functions \(E_1\) and \(E_2\) are normalized by$$\begin{aligned} -2 E_1(0) = -2 E_2(0) = 1. \end{aligned}$$
- (ii)The de Branges spaces \(\mathcal {B}(E_1)\) and \(\mathcal {B}(E_2)\) are both isometrically embedded in the space \(L^2({\mathbb {R}};\mu )\), where the Borel measure \(\mu \) on \({\mathbb {R}}\) is given byand \(\delta _z\) denotes the unit Dirac measure centered at z.$$\begin{aligned} \mu = \pi \delta _0 + \pi \sum _{\lambda \in \sigma } \frac{|\eta (\lambda )|}{|\lambda W'(\lambda )|} \delta _\lambda \end{aligned}$$
- (iii)The corresponding reproducing kernels \(K_1\) and \(K_2\) satisfy the inequality$$\begin{aligned} 2\pi K_2(0,0) \ge 1 \ge 2\pi K_1(0,0). \end{aligned}$$
- (iv)The space \(\mathcal {B}(E_1)\) is a closed subspace of \(\mathcal {B}(E_2)\) with codimension at most one. If \(\mathcal {B}(E_1)\) coincides with \(\mathcal {B}(E_2)\), thenOtherwise, when \(\mathcal {B}(E_1)\) has codimension one in \(\mathcal {B}(E_2)\), we have$$\begin{aligned} \Phi _+(z) = 2\pi K_1(0,z) = 2\pi K_2(0,z), \quad z\in {\mathbb {C}}. \end{aligned}$$where \(\Theta \) is any nontrivial function in \(\mathcal {B}(E_2)\) that is orthogonal to \(\mathcal {B}(E_1)\).$$\begin{aligned} \Phi _+(z)&= 2\pi K_1(0,z) + \Theta (z) \frac{1-2\pi K_1(0,0)}{\Theta (0)} \\&= 2\pi K_2(0,z) - \Theta (z) \frac{2\pi K_2(0,0)-1}{\Theta (0)}, \quad z\in {\mathbb {C}}, \end{aligned}$$
- (i)The de Branges function \(E_0\) is normalized by$$\begin{aligned} -2 E_0(0) = 1. \end{aligned}$$
- (ii)
The de Branges space \(\mathcal {B}(E_0)\) is isometrically embedded in the space \(L^2({\mathbb {R}};\mu )\).
- (iii)The corresponding reproducing kernel \(K_0\) satisfies the inequality$$\begin{aligned} 2\pi K_0(0,0) \ge 1. \end{aligned}$$
- (iv)The space \(\mathcal {B}(E_0)\) is one-dimensional and$$\begin{aligned} \Phi _+(z) = \frac{K_0(0,z)}{K_0(0,0)}, \quad z\in {\mathbb {C}}. \end{aligned}$$
Proof
Under the imposed conditions, all zeros of the functions \(\Phi _-\) and \(\Phi _+\) are simple. Indeed, if some \(\lambda \) was a multiple zero of \(\Phi _-\) or \(\Phi _+\), then \(\lambda \) would have to be a zero of the function W as well since the function in (3) is a Herglotz–Nevanlinna function. As this means that \(\lambda \) belongs to the set \(\sigma \), the coupling condition would then imply that \(\lambda \) is a zero of both functions, \(\Phi _-\) and \(\Phi _+\), so that the function in the numerator of (3) would have a zero of order greater than two at \(\lambda \), which constitutes a contradiction.
This auxiliary result in conjunction with a variant of de Branges’ subspace ordering theorem [25] allows us to verify the uniqueness part of our theorem.
Proof of uniqueness
Case 1: the functions\(\Phi _+^\times \) and \(\Phi _+^\circ \)are both constant. The claim is obvious under these conditions since both functions are equal to one.
We will require the following useful fact about rational Herglotz–Nevanlinna functions in order to establish the existence of solutions to the coupling problem.
Lemma B
Proof
Proof of existence
Let \(\eta \in \hat{{\mathbb {R}}}^\sigma \) be admissible coupling constants. We will establish the existence of solutions to the coupling problem with data \(\eta \) in three steps:
It only remains to verify that solutions depend continuously on the given data.
Proof of stability
Footnotes
- 1.
To be precise, this condition has to be read as \(\Phi _+(\lambda )=0\) whenever \(\eta (\lambda ) = \infty \).
Notes
Acknowledgements
Open access funding provided by University of Vienna.
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