# Alexander Weinstein's *Accademia dei Lincei* obituary

Gaetano Fichera published a commemorative speech for Alexander Weinstein: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Serie 870 (5) (1981), 233-240. He delivered this speech at a Commemoration held in the session of the Accademia Nazionale dei Lincei held on 9 May 1981. We have translated the Italian into English and present a version below.

**Alexander Weinstein, by Gaetano Fichera.**

Alexander Weinstein was born in Saratov, Russia, in 1897. He completed his middle school in Astrakhan, but, when he was still a teenager, the family had to leave Russia and emigrate to Germany, where Weinstein completed his studies, in Würzburg, first, and later in Göttingen. He then moved to Zurich and, with the intention of becoming an astronomer, he worked at the Federal Observatory of that city, making astronomical observations. But he soon realised that his true vocation was directed towards mathematics and, dedicating himself to this, he obtained, under the guidance of Hermann Weyl, a doctorate in mathematics at the University of Zurich in 1921. His thesis, which studied the tensor calculus and linear groups of matrices, was published in Mathematische Zeitschrift.

Weyl, realising the unusual talent of his pupil, endeavoured to put him in contact with some of the greatest mathematicians of the time. Weinstein thus spent a year with Leon Lichtenstein in Leipzig, in 1922. Subsequently Weyl proposed him for a

*Rockefeller International Fellowship*, which allowed Weinstein to spend two years in Rome (1926-27) and to work under the guidance of Tullio Levi-Civita. These had a decisive influence on him, especially as regards his research on hydrodynamics.

Back in Zurich, he obtained the qualification of privatdocent in the Weyl chair. Later he had teaching positions at the universities of Hamburg and Wroclaw. He was already in talks with Albert Einstein for a position as his collaborator in Berlin, when the advent of the Nazis coming to power and the beginning of the persecutions against the Jews, forced Weinstein, like many other scientists in Germany, to emigrate abroad. Weinstein moved to France, where he worked at the Sorbonne and the Collège de France, especially in the context of Jacques Hadamard's seminary. In order to facilitate achieving a permanent university position in France, he presented himself, in 1937, as a candidate for the Doctorate in Mathematical Sciences at the Sorbonne, a title which, as an already well-established scholar, he obtained easily and brilliantly. He also spent a few semesters in England, working at the universities of Cambridge and London.

Back in France, he was forced, in 1940, when the Nazi armies were about to invade that country, to take refuge in America, where he wandered for a long time before finding a definitive place to settle. This was, at that time, a difficult objective, given the overseas flow of numerous European scientists, forced to flee their countries due to Nazi violence and the racial laws that Hitler, as unfortunately we know, had also imposed on the occupied countries.

In the years between 1940 and 1948 Weinstein subsequently worked in the French Free University of New York, in the research group of Harvard University led by Garrett Birkhoff, in the University of Toronto, in the Carnegie Institute of Technology of Pittsburgh, in the Laboratory of Naval artillery in Maryland. Finally, in 1948, he obtained a permanent position at the University of Maryland, where he, together with Monroe Martin, founded the Institute of Fluid Dynamics and Applied Mathematics, which for over fifteen years was one of the major centres in the United States for mathematical analysis and its applications. During this period he helped several impressive young men into scientific research, who later became professors in various universities around the world. Among them I want to explicitly mention: Joe Diaz, Larry Payne, Hans Weinberger, Melvin Lieberstein, Norman Bazley, David Fox, Robert Carroll, and Richard Weinacht. But many other scholars were directly or indirectly influenced by Weinstein's work, including those who have the honour of speaking to you today.

Leaving his position as Research Professor at the University of Maryland in 1967 due to age limits, Weinstein continued his career at the American University in Washington D.C. for a year, and then, from 1968 to 1972, at Georgetown University in the same city. After reaching 75 years of age, he left universities for good.

Among the various awards he received during his lifetime, he always considered his election as Foreign Fellow of our Academy [Accademia Nazionale dei Lincei] in 1964 to be the greatest. In the study of the apartment he occupied in Silver Spring, Maryland, he held the place of honour a painting depicting Galileo Galilei in the Accademia dei Lincei.

In 1957, on the occasion of his 60th birthday, a volume of mathematical writings, originating from a conference held at the University of Maryland, was dedicated to him [

*Proceedings of the Conference on Differential Equations*(University of Maryland, 1965), edited by J B Diaz and L E Payne]. S Gould's book on variational methods in eigenvalue theory is also dedicated to him [

*Variational Methods for Eigenvalue Problems: An Introduction to the Weinstein Method of Intermediate Problems*(1957)].

When he turned eighty, his pupil Joe Diaz edited the publication of a "Selecta" of his works by the publisher Pitman in London.

Weinstein passed away on 6 November 1979, following a surgical operation a few days earlier in a Washington D.C. hospital. He left his wife Marianne, to whom he was bound by deep affection, and who was perhaps the only person who could have any influence on that singular character. He left no children.

Weinstein's scientific production consists of over one hundred publications, including Notes and Memoirs and two Monographs: one published in France in the Mémorial des Sciences Mathématiques [

*Etudes des spectres des équations aux dérivées partielles*(1937)] and one consisting of a volume, written in collaboration with one of his students, published by the Academic Press [A Weinstein and W Stenger,

*Intermediate Problems for Eigenvalues*(1972)].

I like to recall that several of his most significant results were published in Italian journals and, in particular, in our Rendiconti Accademia dei Lincei.

Weinstein's fields of interest can roughly be divided into four areas. We will give each a brief mention here.

**1 - Group theory.**

His early works belongs to this area, including his Doctoral Thesis in Zurich. They concern the representation of linear groups of matrices. Élie Cartan [E Cartan,

*Le principe de dualité e la théorie des groupes simples et semi-simples*(1925)] and Issai Schur [I Schur,*Ueber die Darstellungen der allgemeinen linearen Gruppen*(1928)] were interested in his results.**2 - Hydrodynamics and Applied Mathematics.**

Weinstein dedicated several of his works to the theoretical problems of hydrodynamics. Among these, his important research on the existence of a liquid jet due to the two-dimensional and irrotational flow of a fluid coming out of an arbitrary shaped nozzle deserves to be mentioned. This problem, posed by Helmholtz in 1868, and which analytically translates into a difficult problem of a free boundary, had long remained unsolved. Weinstein, using an ingenious method called "continuity," succeeded in resolving it completely in 1929. He is universally considered the initiator of the theory of existence in this field [see, for example, R Courant and D Hilbert,

*Methods of Mathematical Physics*, vol. II; D Gilbarg,*Jets and cavities*; and M A Lavrentiev,*Variational Methods for Boundary Value Problems*]. His method formed the beginning for the topological-functional procedures later developed by Leray and Schauder. Another well-known result of Weinstein in hydrodynamics is that relating to the solitary wave velocity problem, considered in a 1926 Note in the Accademia dei Lincei and translating into a non-linear problem for a pair of conjugate harmonic functions. The theoretical results obtained by Weinstein were later confirmed by experimental tests performed at the M.I.T. and Johns Hopkins University. Weinstein devoted himself to numerous other problems of hydrodynamics and, more generally, of applied mathematics, obtaining significant results that it would take too long to relate here. I will limit myself only to mentioning the one relating to the mathematical theory of elasticity, concerning the identity between the centre of torsion and the centre of bending in the deformation of an elastic cylinder, and those concerning the spherical pendulum, later taken up by various mechanicians in the study gyroscopes.**3 - Partial differential equations.**

Weinstein's researches relating to the theory of partial differential equations are intertwined with those of applied mathematics and the theory of eigenvalues, therefore they often have an random character, even if the results achieved always demonstrate the originality and remarkable talent of their author. But Weinstein and his students systematically dealt with two particular equations, constructing sufficiently complete theories of them. The first of these is the so-called

(1) $\large \sum_{h=1}^n \Large\frac {\partial^2 u} {\partial x_h^2}\normalsize = \Large\frac {\partial^2 u} {\partial t^2}\normalsize + \Large \frac k t \frac {\partial u}{\partial t}$.

The other is the equation

(2) $\large \sum_{h=1}^n \Large\frac {\partial^2 u} {\partial x_h^2}\normalsize = -\Large\frac {\partial^2 u} {\partial t^2}\normalsize - \Large \frac k t \frac {\partial u}{\partial t}$.

The second of these, if $k$ is a positive integer, is nothing other than the equation of potential functions in $(k + 2)$-dimensional space, with axial symmetry. In this case (2) had been thoroughly studied by Eugenio Beltrami in 1880. Weinstein considers (1) and (2) for arbitrary values of the parameter; for this reason he calls the theory related to (2) Generalised Axially Symmetric Potential Theory (GASPT). He and his pupils obtain numerous interesting results for the two equations and highlight the links between them and other famous equations, such as that of Tricomi, which in the elliptic half plane is a particular case of (2) assuming $n = 1, k = \large\frac{1}{3}\normalsize$ and making an appropriate change of variables, and in the hyperbolic one we obtain from (1) for $n = 1, k = \large\frac{1}{3}\normalsize$ and with a similar change of variables.

*Euler-Poisson-Darboux equation*(the EPD equation) in $n + 1$ variables(1) $\large \sum_{h=1}^n \Large\frac {\partial^2 u} {\partial x_h^2}\normalsize = \Large\frac {\partial^2 u} {\partial t^2}\normalsize + \Large \frac k t \frac {\partial u}{\partial t}$.

The other is the equation

(2) $\large \sum_{h=1}^n \Large\frac {\partial^2 u} {\partial x_h^2}\normalsize = -\Large\frac {\partial^2 u} {\partial t^2}\normalsize - \Large \frac k t \frac {\partial u}{\partial t}$.

The second of these, if $k$ is a positive integer, is nothing other than the equation of potential functions in $(k + 2)$-dimensional space, with axial symmetry. In this case (2) had been thoroughly studied by Eugenio Beltrami in 1880. Weinstein considers (1) and (2) for arbitrary values of the parameter; for this reason he calls the theory related to (2) Generalised Axially Symmetric Potential Theory (GASPT). He and his pupils obtain numerous interesting results for the two equations and highlight the links between them and other famous equations, such as that of Tricomi, which in the elliptic half plane is a particular case of (2) assuming $n = 1, k = \large\frac{1}{3}\normalsize$ and making an appropriate change of variables, and in the hyperbolic one we obtain from (1) for $n = 1, k = \large\frac{1}{3}\normalsize$ and with a similar change of variables.

**4 - Spectral analysis and eigenvalue calculation.**

It is in this area that Weinstein obtained his most outstanding results. He attacked the problem of constructing approximations from below of the natural frequencies of an elastic system, for example a plate, variously attached to the boundary. Already in 1909 Walter Ritz had been able to provide values from above for these frequencies, using the now famous method called the Rayleigh-Ritz method. It is much more difficult to limit these frequencies (eigenvalues) from below. Weinstein, starting from the observation that the Rayleigh-Ritz method can be interpreted as a finite perturbation method starting from an operator (the identically null operator) whose spectral resolution is known, proposed an analogous method for the calculation of eigenvalues from below, that is a method obtained by means of finite perturbations of an appropriate operator of known spectral resolution, which he called the

Weinstein's method was then extended and placed on very general foundations by Weinstein himself and others and, albeit with the serious limitation consisting in assuming a basic operator is spectrally known, it constitutes a brilliant contribution to the theory of eigenvalues. Among the applications to which it has given rise, we only want to recall the one made by N Bazley, a pupil of Weinstein, who, in 1959, managed to lower, with that method, the first two eigenvalues of the helium atom with such accuracy that the approximations obtained practically coincide with the experimental measurements. And Wigner, in his 1959 "Courant lecture" called this result:

From the biographical notes I gave earlier about Weinstein, it appears that he travelled extensively during his life, residing, even for long periods, in different countries and coming into contact and working with many of the greatest mathematicians of his time. This helped to make the field of his scientific interests and his mathematical culture particularly broad. Weinstein, as a mathematician, was certainly "a citizen of the world." But even outside of mathematics his knowledge, especially artistic and literary, was very extensive. He was fluent in five languages: Russian, English, German, French, and Italian, and was, when in a good mood, a brilliant conversationalist with a remarkable sense of humour.

He loved Italy very much, where, as long as he could travel, he spent the summer holidays with his wife. As a true connoisseur, he greatly appreciated the artistic beauties of our country, often unknown to ordinary tourists and to many of the Italians themselves. Rome, a city he knew very well, always remained one of his favourite destinations.

But his traveling a lot, learning to speak many languages, approaching mathematicians from all over the world were not always due to his free choices. From the events of his life it emerges, in a dramatic way, that several times, since his adolescence, he had to earn the right to freedom and to survival itself, with escape and exile. And it is for this reason that, in evaluating, in addition to the scientific work, the human figure, one must have a particular sense of understanding. In fact, the very difficult conditions in which his youth and the first years of his maturity took place, had engraved indelible marks on his personality. He had been constantly forced to move from one country to another, always looking for a position that corresponded to his undoubted merits as a scientist; he had been obliged, countless times, to start all over again, in ever new and very different environments, striving to regain credibility and success; he had had to submit to the judgment of others at an age when a scientist of his calibre had the full right to independence and economic security. All this had ended up disposing his soul to a profound pessimism and creating a complex of insecurity from which he was no longer able to free himself, even when, by now internationally affirmed, he had reached, at the University of Maryland, a safe and considerable prestige. It is only by thinking of all this that one can understand the intimate human nature of this man, tormented and complex, and explain the difficulty that he generally had in human relationships and that made him end his life in almost complete isolation, only with his seriously ill wife, whom he, already advanced in years and of no longer healthy health, lovingly assisted. But the human qualities of a scholar, the sad or happy events of his life, his strengths and limitations as a living creature, are all aspects destined to fade and be forgotten over the years. What really matters and remains is the contribution made to science: and I have no doubts in affirming that the scientific work of Alexander Weinstein has all the characteristics of those destined to last over time.

*basic operator*. He showed that, for a wedged plate, the basic operator can be taken as the one relating to the membrane of the same shape, fixed to the edge. This allowed him, in 1936, to calculate values from below for the first three eigenvalues of the square plate fixed along its edges. A truly exceptional undertaking, at the time when it was accomplished, when the advent of electronic computers was still far away.Weinstein's method was then extended and placed on very general foundations by Weinstein himself and others and, albeit with the serious limitation consisting in assuming a basic operator is spectrally known, it constitutes a brilliant contribution to the theory of eigenvalues. Among the applications to which it has given rise, we only want to recall the one made by N Bazley, a pupil of Weinstein, who, in 1959, managed to lower, with that method, the first two eigenvalues of the helium atom with such accuracy that the approximations obtained practically coincide with the experimental measurements. And Wigner, in his 1959 "Courant lecture" called this result:

*the most striking miracle that has occurred in the course of the development of elementary quantum mechanics*[E P Wigner,*The Unreasonable Effectiveness of Mathematics in the Natural Sciences*(1960)], given that, as he states, the mathematical approach to the physical problem did not allow for such an adherence to experimental reality.From the biographical notes I gave earlier about Weinstein, it appears that he travelled extensively during his life, residing, even for long periods, in different countries and coming into contact and working with many of the greatest mathematicians of his time. This helped to make the field of his scientific interests and his mathematical culture particularly broad. Weinstein, as a mathematician, was certainly "a citizen of the world." But even outside of mathematics his knowledge, especially artistic and literary, was very extensive. He was fluent in five languages: Russian, English, German, French, and Italian, and was, when in a good mood, a brilliant conversationalist with a remarkable sense of humour.

He loved Italy very much, where, as long as he could travel, he spent the summer holidays with his wife. As a true connoisseur, he greatly appreciated the artistic beauties of our country, often unknown to ordinary tourists and to many of the Italians themselves. Rome, a city he knew very well, always remained one of his favourite destinations.

But his traveling a lot, learning to speak many languages, approaching mathematicians from all over the world were not always due to his free choices. From the events of his life it emerges, in a dramatic way, that several times, since his adolescence, he had to earn the right to freedom and to survival itself, with escape and exile. And it is for this reason that, in evaluating, in addition to the scientific work, the human figure, one must have a particular sense of understanding. In fact, the very difficult conditions in which his youth and the first years of his maturity took place, had engraved indelible marks on his personality. He had been constantly forced to move from one country to another, always looking for a position that corresponded to his undoubted merits as a scientist; he had been obliged, countless times, to start all over again, in ever new and very different environments, striving to regain credibility and success; he had had to submit to the judgment of others at an age when a scientist of his calibre had the full right to independence and economic security. All this had ended up disposing his soul to a profound pessimism and creating a complex of insecurity from which he was no longer able to free himself, even when, by now internationally affirmed, he had reached, at the University of Maryland, a safe and considerable prestige. It is only by thinking of all this that one can understand the intimate human nature of this man, tormented and complex, and explain the difficulty that he generally had in human relationships and that made him end his life in almost complete isolation, only with his seriously ill wife, whom he, already advanced in years and of no longer healthy health, lovingly assisted. But the human qualities of a scholar, the sad or happy events of his life, his strengths and limitations as a living creature, are all aspects destined to fade and be forgotten over the years. What really matters and remains is the contribution made to science: and I have no doubts in affirming that the scientific work of Alexander Weinstein has all the characteristics of those destined to last over time.

Last Updated January 2021