On characteristic matrix of Weyl–Titchmarsh type for differential-operator equations, which contains spectral parameter in linear or Nevanlinna's manner

V. I. Khrabustovskii

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 10 (2003) no. 2, pp. 205-227

Résumé

In Hilbert space we consider on finite or infinite interval $(a, b)$ Hamiltonian differential-operator equation, which contains the spectral parameter $\lambda $ in Nevanlinna's manner. For this equation we define the characteristic operator $M(\lambda)$ and prove its existense. We descript $M(\lambda)$, which corresponds to separate bound conditinon, and found the connection between characteristic operators on $(a, b)$, $(a, c)$, $(c, b)$, where $a<c<b$. As application we prove for Sturm-Liouville equation with operator-valued potential the analog of F. S. Rofe-Beketov theorem about reductions of inverse problem on the axis to inverse problems on half-axises. In matrix case, when equation contains $\lambda$ in linear manner and its coefficients are periodic with different periods on half-axises, we find the absolutely continuous part of spectral matrix. The most of results are new even for matrix case and for the case, when equation contains $\lambda$ in linear manner.