# Leonid Andreevich Pastur

### Born: 21 August 1937 in Vinnytsya, Ukraine

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**Leonid Andreevich Pastur**was born in Vinnytsya, Ukraine, a city in west-central Ukraine also known as Vinnitsa. The city had a one time been part of Poland and at another time part of Russia. He was born at an extremely difficulty time when around 10,000 citizens of the city were massacred by the Soviet secret police. He attended secondary school in Mariupol, a city in the south east of the Ukraine close to the Sea of Azov, graduating in 1955. He then went to Kharkov (now known as Kharkiv) where he entered the Faculty of Physics and Engineering at the Kharkov Polytechnic Institute. The authors of [6] write:-

He was an undergraduate at the Kharkov Polytechnic Institute until 1961 when he graduated and began undertaking research at the Institute for Low Temperature Physics and Engineering in Karkov. This Institute was part of the Academy of Sciences of Ukraine, and Pastur began work in the Department of Mathematical Physics which was headed by Vladimir Aleksandrovich Marchenko. We note that Marchenko had just been appointed as Head of the Department when Pastur began research, and Evgenii Yakovlevich Khruslov began research as Marchenko's student at the same time. Pastur was, at this time, appointed as a Junior Research Fellow. In 1964 he was awarded a Candidate's Degree (equivalent to a Ph.D.) in theoretical mathematical physics for his thesisAt that time, the level of teaching of mathematics and theoretical physics in the faculty was extremely high, and this inspired students to study these subjects in more depth. This determined both Pastur's subsequent scientific interests and his style of work. The characteristics of his research activity have a natural connection with theoretical physics, the ability to distil new mathematical questions and promising lines of approach from a physical problem, and the effective application of modern mathematical methods when solving physical problems.

*The dislocation theory of twins in bounded crystals*. In this thesis he studied questions of the dislocation theory of twinning in the theory of strength of materials.

Ilya Mikhailovich Lifshitz (1917-1982), the younger brother of Evgenii Mikhailovich Lifshitz, began working with Pastur in 1963. They developed a theory of disordered systems, a new field of solid state physics having to do with substances that have no large scale structural order. Examples of such materials are amorphous metals, semiconductors, polymers, and many others. Their work on this topic over the following fifteen years was written up in several papers but put together in the monograph

*Introduction to the theory of disordered systems*(Russian) (1982). Andrzej Zardecki writes in a review:-

Pastur was promoted to Senior Research Fellow at the Institute for Low Temperature Physics and Engineering in 1968 and awarded his Doctor of Science degree (equivalent in standard to a D.Sc. of the habilitation) in 1975 for his thesisThis book is devoted to one-particle approximation in the theory of disordered systems. The authors start with a general discussion, exemplified by such properties as the existence of spectral bounds, density of a discrete spectrum and localization. About one-half of the book deals with one-dimensional systems. The concept of the density of states is thoroughly illuminated with model calculations. This is followed by a study of more complex features of disordered systems including the localization length and kinetic properties. In the three-dimensional case, the authors investigate the behaviour of the density of states and of the wave function in the vicinity of the fluctuating spectral bounds. For substitutional disorder a modified perturbation theory, based on the smallness of concentration, is established. Both formal properties of the perturbation series and the applications to the evaluation of the spectrum in the presence of impurity centres are discussed. In the last part of the book, the authors study the penetration of a stream of particles through a disordered layer. This text give a comprehensive treatment of the theory of disordered systems; the bibliography includes work done in the field up to1980.

*Problems in the theory of disordered systems*. The authors of [10] give the following acknowledgement in their paper:-

Since the above quote mentions two important monographs co-authored by Pastur, we should give a few further details of the second at this point. He wroteIt is a pleasure to dedicate this survey to Leonid A Pastur - one of the founding fathers of the rigorous theory of disordered systems. He is a mathematical physicist who masterly knows how to convert physical intuition into mathematical theorems and vice versa. Many of his contributions to the theory of random Schrödinger operators have been ground breaking. Here we only mention the early papers[(with M M Benderskij)'On the spectrum of the one-dimensional Schrödinger equation with a random potential'(1970), 'On the Schrödinger equation with a random potential'(1971), 'On the distribution of the eigenvalues of the Schrödinger equation with a random potential'(1972), 'Behaviour of some Wiener integrals as t → infinity and the density of states of Schrödinger equations with random potential'(1977),(with I Ya Goldsheid and S Molchanov)'A pure point spectrum of the stochastic one-dimensional Schrödinger operator'(1977), and 'Spectral properties of disordered systems in the one-body approximation'(1980)], his survey articles['Spectra of random self-adjoint operators'(1973), and 'Spectral properties of random self-adjoint operators and matrices(a survey)'(1999)]and the monographs[(with I M Lifshits and S A Gredeskul)'Introduction to the theory of disordered systems'(1982), and(with A Figotin)'Spectra of random and almost-periodic operators'(1992)].

*Spectra of random and almost-periodic operators*in collaboration with Alexander Figotin. Russell A Johnson begins a review of the monograph by writing:-

Let us give details of Pastur's career at the Institute for Low Temperature Physics and Engineering in Kharkov. He was Head of the Department of Statistical Methods in Mathematical Physics from 1985 to 1997, and Deputy Director and Head of the Mathematical Division from 1987 to 1997. He served as a part-time Professor at Kharkov State University from 1978 to 1994, and then as Professor in the Department of Mathematics at the University Paris 7, Denis Diderot, from 1995 to 2004. These university appointments, however, did not affect his main appointment at the Institute for Low Temperature Physics and Engineering in Kharkov where he has been employed throughout his career. In Paris, Pastur had other appointments, namely as Head of Department of Theoretical Physics at the University Paris 7, Denis Diderot, from 2003.The present book studies random differential and difference operators. The most important example is the random Schrödinger operator ... These operators have been much studied in recent years ...

Pastur also plays a major role in the administration of research in the Ukraine. For instance he served as Chairman of the Mathematics Panel of the Ukrainian Fund of Fundamental Researches in Kiev from 1992 to 1996, and as Chairman of the Ukrainian Fund of Fundamental Researches in Kiev from 1994 to 1996. At an international level, he was a member of the Executive Committee of the International Association of the Mathematical Physics from 1997 to 2003. We should also note that he has served on the editorial boards of many journals: the International Journal of Low Temperature Physics (1994-); the Journal of Statistical Physics (1987-1989, 1993-1996); Mathematical Physics, Analysis and Geometry (1994-); the Ukrainian Mathematical Journal (1992-); Selecta Matematica Sovetica (1988-1995); Random Operators and Stochastic Equations (1992-); Markov Processes and Related Fields (1999-); and Geometrical and Functional Analysis (2001-).

The most recent monograph by Pastur is

*Eigenvalue distribution of large random matrices*(2011) written in collaboration with Mariya Shcherbina. Terence Tao has reviewed the book and writes:-

Pastur has been honoured for his many contributions. In particular he had been awarded the Ukrainian State Prize in Science and Technology (1985) for his bookThis text is another welcome and modern addition to the literature on the asymptotic spectral statistics of large random matrices. The focus in this text is on the classical invariant Hermitian or symmetric random matrix ensembles, such as the Gaussian Unitary Ensemble, Gaussian Orthogonal Ensemble and Wishart ensemble. However, some space is also devoted to more general classes of ensembles, such as invariant ensembles given by a potential function, or Wigner matrix models in which the upper-triangular entries are jointly independent. Some invariant unitary matrix models are also considered.

*Introduction to the theory of disordered systems*(Russian) written with Ilya Mikhailovich Lifshitz, and elected an Academician of the National Academy of Sciences of Ukraine (1990). He has been an invited speaker at many major conferences including as a special session speaker at the International Congress of Mathematicians held in Berkeley in August 1986, a similar role at the International Congress of Mathematical Physics held in Swansea in 1988, a plenary lecturer at the International Congress of Mathematical Physics held in Leipzig in 1991, and a special session speaker at the II European Congress of Mathematicians held in Budapest in 1996. In May 20013 the conference 'Mathematical Physics of Disordered Systems' was held in Hagen, Germany, in Pastur's honour. The organisers wrote:-

Let us end with the following comments by the authors of [6]:-The Wednesday will be a special day: all sessions will be in honour of Professor Leonid Pastur. The morning session will take place at the usual conference location. To celebrate Leonid's75^{th}birthday the Wednesday afternoon session will take place at the medieval castle Hohenlimburg near Hagen. The conference dinner on Wednesday evening will also takes place at the castle.

The authors of [4] make similar comments:-His vast erudition, both in mathematics and physics, together with his open character and helpful attitude, have always attracted numerous students to him. Many of them now hold doctorates or higher doctorates.

The scope of his research interests is very broad. He continuously follows all the new advances not only in areas close to his own, but also in other areas of mathematics. His immense erudition in mathematics as well as physics, in combination with his sociable disposition and benevolence, has attracted to him numerous students who are now successfully working in Ukraine and in other countries.

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

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