# Aleksandr Yakovlevich Povzner

### Quick Info

Poltava, Ukraine

Washington, D.C., USA

**Aleksandr Povzner**was a Ukranian-born mathematician who worked in the spectral theory of differential operators.

### Biography

**Aleksandr Yakovlevich Povzner**was born into a Jewish family in Poltava, in central Ukraine. At this time Poltava had a large Jewish community of which the Povzner family were a part. Aleksandr was brought up in Poltava where he attended a seven-year school and then a school for metal workers. He then entered Kharkov State University, graduating in 1936. He remained at Kharkov State University where he undertook research, but he was most strongly influenced by Nikolai Grigorievich Chebotaryov who was a professor at Kazan State University. Povzner spent much time with Chebotaryov in Kazan while undertaking research in algebra. He published

*The representation of smallest degree isomorphic to a given Abelian group*(1937), written to give his partial solution to a problem stated by Otto Yulevich Schmidt in his book

*The abstract theory of groups*, namely given an abstract group, find a permutation representation of least degree. Povzner solved this in the case of abelian groups in his first published paper. Chebotaryov gave Povzner and interest in Lie groups and he published

*On binomial Lie groups*(1938) which gave a system of axioms defining a binomial Lie group. This topic provided the material for his Master's thesis (equivalent to a Ph.D.) which was awarded in 1938.

Following the award of his Master's degree, Povzner was appointed as a lecturer in the Department of Algebra of the University of Kharkov in 1938. Two further papers on algebra were published following his appointment, namely

*On nilpotent Lie groups*(1940) and

*Über positive Funktionen auf einer Abelschen Gruppe*Ⓣ (1940). In the first of these he introduced bases with particular properties and, using these, indicated a proof of Garrett Birkhoff's theorem that a nilpotent Lie group can be represented by nilpotent matrices. At this stage in his career, however, his interests changed from algebra to analysis and his first publication in his new area of research was

*Sur les équations du type de Sturm-Liouville et les fonctions "positives"*Ⓣ (1944). However, these years were also the time of World War II and Povzner was conscripted into the Soviet Army in July 1941 serving until 1946. This did not prevent him continuing to undertake research and he published

*On equations of the Sturm-Liouville type on a semi-axis*(1946) and a joint paper with Boris Moiseevich Levitan in the same year entitled

*Differential equations of the Sturm-Liouville type on the semi-axis and Plancherel's theorem*. Levitan, who was one year older than Povzner, had been both an undergraduate and a graduate student at the University of Kharkov before being appointed as a professor in 1941. Both Levitan and Povzner were serving in the army when they collaborated on their joint paper.

After being demobilised from the army, Povzner returned to take up his teaching position at the University of Kharkov and submitted his doctoral dissertation (equivalent to the German habilitation) to Moscow State University. The authors of [5] write:-

Povzner's thesis had far-reaching implications. It was one of the first works in which Gelfand's theory of normed rings and Delsarte and Levitan's theory of generalized displacement operators found a very fruitful application in the spectral analysis of specific operators.After his appointment to the University of Kharkov in 1946, Povzner continued to teach there until 1960 when he took up a position in the Institute of Chemical Physics of the USSR Academy of Sciences. The years during which he worked at Kharkov were ones during which the mathematical school there blossomed due not only to Povzner but also to Naum Il'ich Akhiezer and Vladimir Aleksandrovich Marchenko. Although he continued to work at the USSR Academy of Sciences, in the 1970s he moved to its Institute of Physics of the Earth. He continued to work there when he was over ninety years old.

In addition to the work we have already mentioned, we note that Povzner was the first to apply the technique of transformation operators of Volterra type to spectral theory in 1948. His work on the spectral theory of multidimensional differential operators produced important results on scattering theory for multidimensional Schrödinger operators. He also worked on partial differential equations which describe non-stationary processes. He used computers in an innovative way in solving certain applied mathematics problems and made contributions to statistical physics.

Let us look now at an important book which Povzner wrote in collaboration with V N Bogaevskii, namely

*Algebraic methods in nonlinear perturbation theory*. The book appeared in Russian in 1987 and an English translation was published three years later. The book has five chapters: 1. Matrix theory of perturbations. 2. Systems of ordinary differential equations. 3. Examples. 4. Reconstruction. 5. Partial differential equations. The author's have ambitious aims as explained by James Murdock at the beginning of his review of the English version:-

This promises to be a very important book, although the material presented is not (and is not claimed to be) in definitive form. For a long time, workers in perturbation theory have hoped for a consolidation of all the diverse perturbation methods into a single scheme. There has been a great deal of piecemeal work linking various theories: averaging with normal forms, averaging with multiple scales, matching with multiple scales. One consolidation has been quite successful: Lie theory has largely replaced all other ways of handling near-identity transformations, in both Hamiltonian and non-Hamiltonian settings, for both averaging and normal forms. But nothing has as yet fully bridged the gap between "oscillatory" theories (such as averaging, normal forms, and certain kinds of multiple scales methods) and "transition layer" theories (such as matching and other kinds of multiple scales). It is this which is attempted in the present book. Roughly, the claim seems to be that a generalization of the normal form method, together with a generalized idea of rescaling (which the authors call "reconstruction"), is applicable to all perturbation problems. Matching arises when two (or more) reconstructions are possible. In this case the general process of reconstructing-and-normalizing will already have put the problem in the simplest form so that the correct way of matching becomes clear.As to Povzner's character, the authors of [7] write:-

Povzner was a wonderful lecturer. His lectures and seminars at Kharkov University were always extremely popular and attracted many young people. Undergraduate and graduate students and colleagues remember well the attractiveness of his personality, which exerted a tremendous influence on all those around him. Especially appealing were his unconventional thinking and opinions, not only in questions of science but also in various aspects of culture and modern life.

### References (show)

- Aleksandr Yakovlevich Povzner,
*Methods Funct. Anal. Topology***15**(1) (2009), 1-2. - Aleksandr Yakovlevich Povzner (June 27, 1915-April 22, 2008) (Russian),
*Zh. Mat. Fiz. Anal. Geom.***4**(3) (2008), 441-442. - Aleksandr Yakovlevich Povzner (on the occasion of his ninetieth birthday) (Russian),
*Zh. Mat. Fiz. Anal. Geom.***1**(1) (2005), 140. - Ju M Berezanskii, I M Gelfand, B Ja Levin, V A Marchenko and K V Maslov, Aleksandr Jakovlevich Povzner (on the occasion of his sixtieth birthday) (Russian),
*Uspehi Mat. Nauk***30**(5)(185) (1975), 221-226. - Ju M Berezanskii, I M Gelfand, B Ja Levin, V A Marchenko and K V Maslov, Aleksandr Jakovlevich Povzner (on the occasion of his sixtieth birthday),
*Russian Math. Surveys***30**(5) (1975), 177-183. - Yu M Berezanskii, V A Marchenko, K V Maslov, S P Novikov, L A Pastur, F S Rofe-Beketov, Ya G Sinai, L D Faddeev and E Ya Khruslov, Aleksandr Yakovlevich Povzner (Russian),
*Uspekhi Mat. Nauk***63**(4)(382) (2008), 175-176. - Yu M Berezanskii, V A Marchenko, K V Maslov, S P Novikov, L A Pastur, F S Rofe-Beketov, Ya G Sinai, L D Faddeev and E Ya Khruslov, Aleksandr Yakovlevich Povzner,
*Russian Math. Surveys***63**(4) (2008), 771-772.

### Additional Resources (show)

Other websites about Aleksandr Yakovlevich Povzner:

Written by J J O'Connor and E F Robertson

Last Update November 2010

Last Update November 2010