Dynamic inverse problem for Jacobi matrices

Alexandr Mikhaylov; Victor Mikhaylov

doi:10.3934/ipi.2019021

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2019, Volume 13, Issue 3: 431-447. Doi: 10.3934/ipi.2019021

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Dynamic inverse problem for Jacobi matrices

Alexandr Mikhaylov^1,2, , and
Victor Mikhaylov^1,2,

1.
St. Petersburg Department of V.A. Steklov Institute of Mathematics of the Russian, Academy of Sciences, 27, Fontanka, 191023 St. Petersburg, Russia
2.
Saint Petersburg State University, St.Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034, Russia

Received: September 30, 2017

Revised: September 30, 2018

Published: March 2019

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Abstract

We consider the inverse dynamic problem for a dynamical system with discrete time associated with a semi-infinite Jacobi matrix. We derive discrete analogs of Krein equations and answer a question on the characterization of dynamic inverse data. As a consequence we obtain a necessary and sufficient condition for a measure on a real line to be a spectral measure of a semi-infinite discrete Schrödinger operator.

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Mathematics Subject Classification: Primary: 35R30, 93B30; Secondary: 39A12, 35C15, 35P99.

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