arXiv subjects

Nurmukhamet Valeev

Publications and source records attributed to Nurmukhamet Valeev.

2 recordsLinked to original sources

arXiv2024

A new proof of the Perron-Frobeniuos theorem, a variational approach

We develop the Perron-Frobenius theory using a variational approach and extend it to a set of arbitrary matrices, including those that are neither irreducible nor essentially positive, and non-preserved cones. We introduce a new concept called a quasi-eigenvalue of a matrix, which is invariant under orthogonal transformations of variables, and has various useful properties, such as determining the largest value of the real parts of the eigenvalues of a matrix. We extend Weyl's inequality for the eigenvalues to the set of arbitrary matrices and prove the new stability result to the Perron root of irreducible nonnegative matrices under arbitrary perturbations. As well as this, we obtain new types of estimates for the ranges of the sets of eigenvalues and their real parts.

math.AP↗

arXiv2019

On an optimal potential of Schrödinger operator with prescribed $m$ eigenvalue

The purpose of this paper is twofold: firstly, we present a new type of relationship between inverse problems and nonlinear differential equations. Secondly, we introduce a new type of inverse spectral problem, posed as follows: for a priori given potential $V_0$ find the closest function $\hat{V}$ such that $m$ eigenvalues of one-dimensional space Schrodinger operator with potential $\hat{V}$ would coincide with the given values $ E_1 $, $ \ldots $, $ E_m \in \mathbb {R} $. In our main result, we prove the existence of a solution to this problem, and more importantly, we show that such a solution can be directly found by solving a system of nonlinear differential equations.

math.AP↗