Description of Generalized Resolvents and Characteristic Matrices of Differential Operators in Terms of the Boundary Parameter

V. I. Mogilevskii

Matematičeskie zametki, Tome 90 (2011) no. 4, pp. 558-583

Résumé

We supplement and further develop well-known results due to Shtraus on the generalized resolvents and spectral functions of the minimal operator $L_0$ generated by a formally self-adjoint differential expression of even order with operator coefficients given on the interval $[0,b\rangle$, where $b\le\infty$. Our approach is based on the notion of a disintegrating boundary triple, which allows us to establish a relation between the Shtraus method and boundary-value problems with spectral parameter in the boundary condition. In particular, we obtain a parametrization of all the characteristic matrices $\Omega(\lambda)$ of the operator $L_0$ in terms of the spectral parameter corresponding to a boundary-value problem. Such a parametrization is given as a block representation of the matrix $\Omega(\lambda)$, as well as by formulas similar to Krein's well-known formula for generalized resolvents.

Détail

Keywords: differential operator of even order, minimal operator, self-adjoint operator, generalized resolvent, characteristic matrix, boundary-value problem, deficiency index, boundary triple, holomorphic function, Nevanlinna function, Weyl function.