# Vladimir Aleksandrovich Marchenko

### Born: 7 July 1922 in Kharkov, Ukraine

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**Vladimir Aleksandrovich Marchenko**'s parents were Olga Fedorovna and Alexander Grigoryevich Marchenko. Alexander Grigoryevich was from a working class background but had a good education at the St Petersburg Academy of Forestry. He worked at the Novoaleksandriia Institute of Agriculture and Forestry in Poland until 1914 when, due to the outbreak of World War I, it was transferred to Kharkov. He married Olga Fedorovna who was a school teacher from St Petersburg, and they had four children: Irina, Dmitri, Sergei and Vladimir. By the time their youngest son Vladimir was born in 1922, Alexander Grigoryevich was a professor at the Institute in Kharkov. Let us note that Irina went on to become an expert in art history, Dmitri became an engineer, and Sergei studied medicine but was killed in 1944 during World War II. Vladimir Aleksandrovich's childhood was far from easy. In 1929, when he was seven years old, his father was convicted of attempting to prevent massive deforestation which was seen to be necessary to further socialism. Over the next years his father was seldom able to be at home and Alexander Grigoryevich died in 1940. The 1930s were extremely difficult in Kharkov as attempts were made to eradicate Ukrainian culture with arrests and murders of intellectuals.

Despite the difficulties, Vladimir Aleksandrovich had a good school education learning foreign languages and excelling at chemistry which was his favourite subject. Olga Fedorovna was a great source of strength to the family during these difficult years and her efforts gave all the children an excellent start in life. After graduating from high school in 1939, Vladimir Aleksandrovich entered the Department of Physics in Leningrad University. Although World War II started in September 1939 with the German invasion of Poland, this had little effect in Russia and the Ukraine and Marchenko was able to take courses in physics, mechanics and mathematics. His university career progressed well and by the summer of 1941 he had finished two successful years when he returned to his family home in Kharkov for the summer vacation. However, on 22 June 1941 the German armies invaded their former allies pushing rapidly east into Soviet lands. At first their main advance was aimed towards Moscow, but by August they made a strong push in the south deep into the Ukraine capturing Kharkov on 24 October 1941. Marchenko was not called up for military service since he was severely myopic. Little could be done to halt the German advance as their armies approached Kharkov, nor was fleeing a realistic option by this time, so Marchenko remained in Kharkov with his mother, Irina and her daughter, all other members of the family having left earlier. An attempt by the Red Army to retake the city from the Germans failed in May 1942.

The German occupation of Kharkov lasted until 1943. During this time of great hardship Marchenko was able to use his expertise in chemistry to good advantage since he was able to manufacture salt and matches which were is very short supply. His mother sold what he made in exchange for basic food and this allowed them to survive. The first liberation of the city in February 1943 only lasted until March 1943 when again it was captured by the Germans. After the city was finally liberated on 23 August 1943, Marchenko was able to continue his education which he did in the Mathematics Department of the Physics and Mathematics Faculty of Kharkov University. By this time he had decided that mathematics was the topic he wanted to specialise in and he obtained his first degree from Kharkov University in 1945. Continuing his studies at Kharkov he was awarded his candidates degree (equivalent to a Ph.D.), advised by Naum Samoilovich Landkof who had been a student of Mikhail Alekseevich Lavrent'ev, in January 1948 after defending his thesis

*Summation methods for generalized Fourier series*. He was appointed as a lecturer at Kharkov University in 1950 and he submitted his doctoral thesis (equivalent to the German habilitation)

*Some questions in the theory of one-dimensional linear second-order differential operators*in 1951. In addition to Landkof, Marchenko was influenced by other Kharkov mathematicians, particularly Naum Il'ich Akhiezer, an expert on function theory and approximation theory, and Aleksandr Yakovlevich Povzner (1915-2008) who was the first to apply the technique of transformation operators of Volterra type to spectral theory in 1948.

Marchenko was appointed onto the staff in 1950 and worked at Kharkov University as an assistant professor in the Department of the Theory of Functions. In 1952 he was promoted to professor in the Department of Mathematical Physics. He moved to the Department of Computational Mathematics in 1959 where he served as head until 1961. In that year Marchenko was appointed to the Physical-Technical Institute of Low Temperature of the National Academy of Sciences of the Ukraine. This Institute in Kharkov had been founded in 1960 largely through the efforts of Boris Ieremievich Verkin (1919-1990) who had been a friend of Marchenko's since they were children growing up in Kharkov. Discussions between Verkin and Marchenko led to the, perhaps surprising yet very successful, decision to create a mathematics department within the Institute and Marchenko was appointed Head of the Department of Mathematical Physics at the Institute. However, he continued to hold his chair at the University.

We now look at some of his mathematical contributions [9]:-

In the 1950s Marchenko was influenced by ideas of his colleagues Akhiezer and Povzner. He obtained important results in approximation theory, in particular his results concern the approximation theory of almost periodic functions. Also in the 1950s he studied the asymptotic behaviour of the spectral measure and of the spectral function for the Sturm-Liouville equation. In [12] this work is described as follows:-Marchenko's scientific interests generally centre around problems in mathematical analysis, the theory of differential equations, and mathematical physics. At the very start of his scientific activity, he obtained fundamental results that had a major influence on the development of mathematics.

With great success, Marchenko applied his methods to the Schrödinger equation. Later Marchenko published monographs on this work:He is well known for his original results in the spectral theory of differential equations, including the discovery of new methods for the study of the asymptotic behaviour of spectral functions and the convergence expansions in terms of eigenfunctions. He also obtained fundamental results in the theory of inverse problems in spectral analysis for the Sturm-Liouville and more general equations.

*Spectral theory of Strum-Liouville operators*(1972); and

*Sturm-Liouville operators and their applications*(1977). The second of these is essentially a reworked and augmented version of the first. Mamim M Skriganov, reviewing the 1977 monograph writes:-

Earlier, in collaboration with Z S Agranovich, he had writtenThe spectral theory of Sturm-Liouville operators is a classical domain of analysis, comprising a wide variety of problems at the present time. Besides the basic results on the structure of the spectrum and the eigenfunction expansion of regular and singular Sturm-Liouville problems, it is in this domain that one-dimensional quantum scattering theory, inverse spectral problems and, of course, the surprising connections of the theory with nonlinear evolution equations first become related. The goal of the monograph under review is to present these questions from a unified point of view, namely from the point of view of applying the technique of transformation operators of Volterra type. ... The monograph is carefully written, with attention paid to complete details in the proofs. Each chapter contains numerous problems that supplement the basic text. All this makes the book eminently suitable both as a textbook and as a reference work for the specialist.

*The inverse problem of scattering theory*which appeared in English translation in 1963. Paul Roman writes:-

The periodic case of the Korteweg-de Vries equation was solved by Marchenko in 1972. He used the method of the inverse problem in the theory of dissipation. Volodymyr Petryshyn in [12] describes other work of Marchenko including his work on self adjoint differential operators:-Problems where it is required to ascertain the spectral data which determine a differential operator uniquely, and where a method is sought for constructing this operator from the data, are usually called "the inverse problem" of spectral theory. Probably the most important class of such problems is the inverse problem of scattering theory. The present monograph discusses in detail the particular methods developed previously by the same authors to deal with the inverse scattering problem. ... This book is one on pure mathematics and not on physics as such. However, in view of the lucid presentation and excellent pedagogical approach, this volume may be used with great advantage by any theoretical physicist. Incidentally, quite apart from the principal topic, various methods of functional analysis and algebra which are used should also be of great interest to both pure and applied mathematicians.

In addition to the important monographs mentioned above, other major texts written by Marchenko includeMarchenko made significant contributions to the theory of self-adjoint differential operators with infinitely many independent variables and also to the theory of spaces of functions of infinitely many variables as inductive limits of locally convex function spaces.

*Nonlinear equations and operator algebras*(1986). He writes in the Preface to this book:-

In 1992 Marchenko's monographWe systematically present a method for solving some physically important nonlinear equations that is based on the replacement of a given equation by an equation of the same form with respect to functions that take values in an arbitrary operator algebra. The solution of an operator equation in the form of a travelling wave(a one-soliton solution)is elementary. The solutions of the original equation are obtained from the one-soliton operator solutions by bordering them with special finite-dimensional projectors. Arbitrariness in the choice of the operator algebra and the bordering projectors allows us to find broad classes of solutions of the Korteweg-de Vries, Kadomtsev-Petviashvili, nonlinear Schrödinger, sine-Gordon, Toda lattice, Langmuir and other equations. In these classes solutions are contained that can be obtained by the inverse problem method and by the methods of algebraic geometry, and also solutions that do not reduce to these methods.

*Orthogonal functions of a discrete argument and their application in geophysics*was published and in 2005, in collaboration with Evgeni Yakovlevich Khruslov, he wrote

*Homogenization of partial differential equations*.

In addition to his mathematical research we must mention his service as an editor of several journals: he was editor-in-chief of

*The theory of functions, functional analysis, and their applications*for nearly thirty years; an honorary editor of

*Mathematical Physics, Analysis, Geometry*; an editor of the

*Proceedings of the Ukrainian Academy of Sciences*; and an editor of

*Inverse Problems*. The authors of [7] comment on his skills as a teacher:-

Marchenko has received numerous honours. In 1961 he was elected a corresponding member of the Ukrainian Academy of Sciences, and in 1969 he was elected a full member. In 1987 he was elected to the USSR Academy of Sciences. He was awarded the Lenin Prize in 1962 for his series of papers on inverse problems of spectral analysis. He has also been awarded the N M Krylov Prize in 1980, the State Prize of the USSR in 1989, and the N N Bogolyubov Prize in 1996. In 1992 he was awarded the title "Honoured scientist and technologist of the Ukraine". In 1997 he was awarded an honorary doctorate by the Sorbonne in Paris. He was elected a member of the Norwegian Academy of Science and Letters in 2001 and the following year he was awarded the Ukrainian Order of Yaroslav Mudryi. He has served as president of the Kharkov Mathematical Society for a number of years.For several years Marchenko has been engaged in teaching at Kharkov University. He devotes much effort and energy to this work, and his influence on whole generations of Kharkov mathematicians would be hard to overestimate. The characteristics of Vladimir Aleksandrovich's pedagogic activities are a high academic standard and a continuing quest for new methods of instruction. There are several dozen Ph.D's and Doctors of Science among his students.

Outside mathematics, Marchenko is married with a son and a daughter. He has had many hobbies such as skiing, kayaking and stamp collecting. The authors of [3] give the following tribute:-

The breadth of Marchenko's scientific interests and erudition, his limitless dedication to science, high demands on himself, constant attention to students and colleagues, kind-heartedness and readiness to give any amount of help are well known to all who have had the good fortune to meet and work with him.

**Article by:** *J J O'Connor* and *E F Robertson*

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