Matrix finite-zone Dirac-type equations
Consider the problem \[ \frac{dw}{dx}=(izj + \xi(x))w \tag{1} \] on the half-line \((0,\infty)\) and the line \((-\infty ,\infty)\). Here, \(\xi(x)\), \(w(x,z)\) and \(j\) are \(n\times n\)-matrices (with \(n=2m\)) of the form \[ j=\begin{pmatrix} I_m&0\\ 0&-I_m\end{pmatrix},\qquad \xi(x)=\begin{pmatrix} 0&q(x)\\q^*(x)&0\end{pmatrix}, \] where \(q\) is assumed to be locally bounded and \(z\) is the spectral parameter. In the case of the half-line, the boundary condition \(w(0,z)=I_n\) is imposed. In that case, \(\varphi (z)\) is called a Weyl-Titchmarsh matrix function for (1) if \[ \int_0^\infty \begin{pmatrix} \varphi ^*(z)& I_m\end{pmatrix} w^*(x,z)w(x,z) \begin{pmatrix} \varphi (z)\\ I_m\end{pmatrix} dx<\infty . \] It is known that there is a unique Weyl-Titchmarsh matrix function. The case \((-\infty ,\infty)\) is reduced to the half-line case. Assuming that the Weyl-Titchmarsh matrix function has the form \[ R(z)^{-\frac 12}\begin{pmatrix} iP(z)& Q_1(z)\\ Q_2(z)&iS(z)\end{pmatrix} \tag{2} \] with matrix polynomials \(P\), \(Q_1\), \(Q_2\), \(S\), and a scalar polynomial \(R\), it is shown that they have to satisfy certain conditions. Finally, under condition (2) and in the case \(m=1\), a procedure to recover the potential \(q\) from the Weyl-Titchmarsh function is given. In particular, \(q\) is quasiperiodic.
scientific article; zbMATH DE number 1836683
scientific article