Kholkin O. Problems of spectral analysis of self-adjoint and non-self-adjoint differential operators

Dissertation work is devoted to research of connections between spectral and oscillatory properties of differential equations with operator (matrix) coefficients. Properties of spectrum of such differential operators are studied. The oscillatory theorem is proved for the problem of Sturm-Liouville on finite and infinite intervals with bounded operator coefficients in separable Hilbert space of Н. An important special case of this result is Morse Index Theorem. For differential equation of arbitrary order (even as well as odd), with operator coefficients a fundamental solution is constructed with a boundary condition on a singular end. New generalization of Sturm's oscillatory theorem has been received for the self-adjoint differential equations of arbitrary even order with operator-valued coefficients for the boundary condition of general view. Topological interpretation of these theorems is given and their connection with symplectic geometry is examined. For equation with block-triangular, growing at infinity matrix potential is obtained decreasing and growing at infinity matrix solutions. Sufficient conditions at which a spectrum of such non-self-adjoint differential operator is real and discrete are obtained. Green's function is constructed, its series expansion is obtained, Parseval equality is proved. Connection between spectral and oscillatory properties of such operators on an finite interval and semi-axis is set. The sufficient conditions of essential self-adjoint are obtained in L2(En) of an operator determined by formally self-adjoint elliptic differential expression in space vector-functions.