Bitangential direct and inverse problems for systems of integral and differential equations (Q5891747)

The book deals with a system of ordinary differential equations of the form NEWLINE\[NEWLINE Y'(x,\lambda)=i\lambda Y(x,\lambda)H(x)J,\quad 0\leq x<d, \tag{1} NEWLINE\]NEWLINE where \(\lambda\) is the spectral parameter, \(Y(x,\lambda), H(x), J\) are \(m\times m\) matrices, \(H(x)\geq 0,\) and \(J^\ast=J,\) \(J^\ast J=I_m\) (\(I_m\) is the identity matrix). The authors study the so-called inverse monodromy problem (IMP) of recovering \(H(x)\) from the given monodromy matrix \(W(\lambda)\) and artificially constructed chains \(b_1(x,\lambda)\), \(b_1(x,\lambda)\), \(x\in[0,d]\), of matrices connected with system (1). The IMP corresponds to the simplest case when the characteristic numbers are real. Under additional restrictions on the potential, the authors provide the solution of the IMP. Then, the authors consider the so-called inverse input scattering problem (IISP) and inverse input impedance problem (IIIP). These inverse problems are reduced to the IMP. The names IISP and IIIP are not natural since the spectral parts of the data are two particular cases of the Weyl matrices related to different boundary conditions; moreover, the above mentioned chains do not fit to scattering and impedance data.NEWLINENEWLINEThe contents of the book reflects the authors' results published in their recent papers.