Instability indices of differential operators

The instability index of a linear operator with point spectrum is defined to be the total multiplicity of its eigenvalues with positive real parts. Under certain conditions the computation of the instability index of an unbounded nonsymmetric operator acting in a Hilbert space can be reduced to an analogous problem for a certain selfadjoint operator. It is shown that if the original operator is a differential operator acting in a space of vector-valued functions with a single variable, then the selfadjoint operator corresponding to it can be found in the form of an integro-differential operator which can be realized constructively by solving a special elliptic boundary value problem. Analogues of known theorems of Morse on the connection between the instability index and the number of conjugate points are established for this integro-differential operator.
Bibliography: 26 titles.