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https://doi.org/10.36719/2789-6919/40/108-113

Gunay Salmanova

Ganja State University

Doctor of Philosophy (PhD) in Mathematics

gunay-salmanova@mail.ru

https://orcid.org/0009-0002-4502-7349

 

The Multilevel Completeness of the Root Subspaces System in a Two-parameter Problem Dependent on Polynomial Parameter

 

Abstract

The only generally accepted method for solving complex equations dependent on multiple parameters is the method of separation of variables. The investigation of complex equations using this method reduces the problem to the study of systems of ordinary differential equations. This, in turn, necessitates the study of concepts such as eigenvectors, associated vectors, multi-completeness, multi-separation, and others for multiparametric systems.

The study of multiparametric systems obtained through the separation of variables is considered the only viable approach to solving the Cauchy problem for operator-differential equations. For a multiparametric system, a specific equation is associated with the tensor product of the spaces influenced by the individual equations of the system. It is demonstrated that the eigenvectors and associated vectors of this equation coincide with the eigenvectors and associated vectors of the investigated system. The construction of the equation involves utilizing an abstract analogue of the resultant of two polynomial sets. It is worth noting that this method also enables the study of multiparametric systems that are not linearly dependent on the parameters.

This work examines  two-parameter systems that are nonlinearly dependent on the parameters and are not self-adjoint. The multi-completeness and multi-basis properties of the eigen and associated elements of various two-parameter systems are demonstrated. Under certain conditions, the eigen and associated elements of two-parameter systems form a multi-complete system in the tensor product of spaces.

Keywords: operators, eigenvalue, multi parameter system, Hilbert space, two parameter system

 


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