Science Development Foundation Under the President of the Republic of Azerbaijan has announced the list of 16 winning projects for the call for proposals for “Basic Grant Competition”. Two proposals were submitted to this competition by the Department of Mathematics of the Khazar University, both of which are among the winners. One of these proposals is titled “Methods of spectral theory and non-harmonic Fourier analysis in some problems of classical and quantum mechanics”, whose manager is Ali Huseynli, the head of the Department of Mathematics. In the project, along with Lala Atamova (Ph.D), another member of the Department of Mathematics, there participate 6 researchers from Baku State University, Azerbaijan State Oil and Industrial University, Azerbaijan University of Architecture and Construction, Institute of Mathematics and Mechanics of the National Academy of Sciences, one researcher from each organization. The project period is 12 months.

The project is dedicated to the study of some mathematical properties of some equations of Quantum Physics. The role of the Dirac and Schrödinger operators in many areas of physics, especially in quantum mechanics is well known. The various spectral properties of these operators characterize the states of quantum systems and are one of the main problems of modern functional analysis. Various spectral problems for these operators, as well as Schrödinger and other differential operators with operator-valued coefficients are studied by means of modern methods of functional analysis.

The aim of the project is to study spectral problems arising in the mathematical modeling of some concrete problems of classical and quantum mechanics, also to answer some questions of Non-Harmonic Analysis which are directly related to the study of considered spectral problems.

The spectral problem for the Schrödinger operator considered in the project is studied in the grand-Lebesgue spaces. These spaces are newly introduced and are used in the modern theory of function spaces and partial differential equations.

The project is also devoted to the study of some problems in the Theory of Frames, which is one of the modern topics of applied harmonic analysis. It should be noted that Frame Theory and Wavelet Analysis are widely used in the theoretical foundations of modern achievements of signal processing, image recognition, and modern crystallography. From this point of view, the mathematical problems considered in the project are not just of great theoretical importance but are also important in solving many applied problems.

The expected results require extensive use of many areas of mathematics, especially the methods of Spectral Theory, Non-Harmonic Fourier Analysis, and Functional Analysis.