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Analytic and algebraic methods in spectral problems

Work number - M 7 AWARDED

Authors: Goriunov A. S., Karadzhov Yu.A.

Theseriesispresented by the Institute of Mathematics of NAS of Ukraine.

It consists of 16research papers published during 2009-2014.

The results of series belong to one of the central branches of modern mathematical physics concerning spectral problems of quantum mechanics. The main objects of study are the Schrödinger equation and its generalizations.

The method of classification is developed for new types of matrix form-invariant Schrödinger equations, which are not covered by previously known methods. Such problems are very complex and to construct their solutions one needs to use algebraic and analytical methods. In particular, the eigenvalues and the basic states are found and quadratic integrability of perturbed states is proved.

The differential operators of Schrödinger type with distributional potentials, in particular, measures and some of their derivatives, are investigated. A new approach to the analysis of such operators has been developed, it is based on their representation as quasi-differential operators with Shin-Zettl quasi-derivatives. New results regarding regularization of some classes of differential operators with generalized functions as coefficients are obtained. In particular, sufficient conditions of norm resolvent approximation of quasi-differential operators of arbitrary orderby differential operators with smooth coefficients are found. Also, the constructive description of the main classes of extensions (self-adjoint, maximal dissipative, maximal accumulative, real) of minimal operators generated by the quasi-differential expressions is given.

The series consists of 16 papersinleading scientific editions. Of them 8areininternational journals with impact factor. Total scope of papers is184 p.

All publications arepeer-reviewed, indexed by  Google Scholar and count 160 citations, h-index= 7;

10papersareindexedbytheinternationaldatabaseSCOPUS and count in this base 67citations;

12papersareindexedbytheinternationaldatabaseMathSciNet  and count in this base 42citations