For 4 decades, he headed the Mathematical Physics Department at the Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine.
[2] Marchenko made fundamental contributions to the analysis of the Sturm–Liouville operators.
He introduced one of the approaches to the inverse scattering problem for Sturm–Liouville operators, and derived what is now called the Marchenko equation.
Together with Leonid Pastur, Volodymyr Marchenko discovered the Marchenko–Pastur law in random matrix theory.
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