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Direct and inverse spectral problem for a system of differential equations depending rationally on the spectral parameter. (English) Zbl 1020.34024

Here, \(2\times 2\) systems of first-order differential equations of the form \[ \frac{dy}{dx}(x,\lambda)= i\left(\frac{b_1\beta_1^*(x)\beta_1(x)} {\lambda-d_1}+\frac{b_2\beta_2^*(x)\beta_2(x)}{\lambda-d_2}\right) y(x,\lambda), \quad x\geq 0,\;\lambda\in \mathbb{C},\tag{1} \] are studied, where \(b_p=\pm 1\), \(p=1\), \(2\), \(d_1\neq d_2\), are real constants, and the vector functions \(\beta_p\) have the property \(\beta_p(x)\beta_p(x)^*=1\), \(x\geq 0\), \(p=1\), \(2\). Systems of this type are closely related to nonlinear equations, as sine-Gordon equation.
The authors introduce, for system (1), a notion of \(W_p\)-function – \(p\)th Weyl function – defined for spectral parameters in a complex neighbourhood, which depends on \(b_p\), of \(d_p\). These Weyl functions are characterized in terms of the asymptotics of the fundamental solution to system (1).
The first result of the paper concerns the existence and uniqueness of the Weyl solution to system (1), under the additional hypotheses that the first component of \(\beta_p\) is not zero at \(0\), \(p=1\), \(2\), and that \(\beta_p\) is absolutely continuous with bounded derivative, \(p=1\), \(2\). This is done using an associated system which is a perturbation of a Dirac-type system, and whose elementary solution is obtained from the elementary solution to (1) by a Bäcklund-Darboux transformation. The main technical result here is a representation for the elementary solution to the auxiliary system.
The inverse problem associated to (1) consists of recovering \(\beta_p\), \(p=1,2\), from the \(W_p\)-functions, when \(b_p\) and \(d_p\), \(p=1,2\), are given. One proves the uniqueness of recovering of \(\beta_p\), if \(\beta_p>0\) and \(\beta_p\) have bounded derivatives. If the latter hypothesis is weakened at requiring \(\sup_{0\leq x\leq l\infty}\|\beta'_p(x)\|\leq C_l\), \(l\geq 0\), then \(\beta_p\), \(p=1,2\), are uniquely recovered form Weyl functions satisfying a certain asymptotic condition.
The last section of the paper contains explicit solutions to the direct problem for (1) in some special cases. The last result in the paper is an partial answer to the more difficult part of the inverse problem, the existence; that is one gives a class of functions that are Weyl functions for systems (1), and the corresponding coefficients \(\beta_p\), \(p=1,2\), are constructed explicitely.

MSC:

34B20 Weyl theory and its generalizations for ordinary differential equations
34B07 Linear boundary value problems for ordinary differential equations with nonlinear dependence on the spectral parameter
34L05 General spectral theory of ordinary differential operators
47E05 General theory of ordinary differential operators
34A55 Inverse problems involving ordinary differential equations
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