Today the regular Institute seminar was held. In the seminar of the Institute of Mathematics and Mechanics, the report doc. ph.-m.s. Bilender Pasha ogly Allahverdiyev will speak with presentation entitled “Some problems of the spectral theory of self-adjoint and non-self-adjoint operators in Hilbert spaces”.

Some problems of spectral theory of non-self-adjoint operators and operator-valued functions with a continuous spectrum acting on Hilbert spaces are studied. A description of all self-adjoint and non-self-adjoint (maximal dissipative, accumulative) extensions of symmetric Schrödinger, singular Sturm-Liouville, Dirac type (Hamiltonian) and difference (infinite Jacobi matrices, discrete Sturm-Liouville and discrete Dirac type (Hamiltonian) operators) is given in terms of boundary conditions. We construct a self-adjoint dilations of the maximal dissipative extensions of these operators and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilations. We establish a functional model of the dissipative operators and construct its characteristic functions in terms of the scattering matrix of the dilations. Completeness of eigenvectors and associated vectors (or root vectors) for these operators are solved. These problems are solved for singular Sturm-Liouville problems and Dirac type (Hamiltonian) systems involving spectral parameter in the boundary conditions, Schrödinger, singular Sturm-Liouville, Dirac type (Hamiltonian) and difference operators (infinite Jacobi matrices, discrete Sturm-Liouville and discrete Dirac type (Hamiltonian) operators) with matrix coefficients. Some problems of spectral analysis for impulsive singular Sturm-Liouville and Dirac type (Hamiltonian) systems are solved. The spectral problems of the Sturm-Liouville and Dirac operators are considered on the time scale. Some problems of spectral theory for regular and singular q-Sturm-Liouville, q-Dirac, q-Hamiltonian, Hahn-Sturm-Liouville, Hahn-Dirac, Hahn-Hamiltonian systems have been solved. Riesz and Bari basicity problems of eigenvectors and associated  vectors (or root vectors) of some differential and q-difference operators have been solved. Existence and uniqueness theorems of nonlinear singular Sturm-Liouville, q-Sturm-Liouville, Hahn-Sturm-Liouville problems have been proved.

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