# Franz Rellich

### Born: 14 September 1906 in Tramin, South Tirol, Austro-Hungarian Empire (now Italy)

Died: 25 September 1955 in Göttingen, Germany

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**Franz Rellich**was born in Tramin, South Tirol, which, at that time was part of Austria. The village is now in Italy (the South Tirol was ceded to Italy in the Treaty of Saint Germain in 1919 following World War I) with its Italian name of Termeno. Rellich entered the Gymnasium in Graz in 1916, studying there until 1924. After graduating from the Gymnasium, he entered the University of Graz in 1924 where he studied mathematics and physics before moving to the Georg-August University of Göttingen in 1926. There he joined a close-knit circle of young mathematicians and physicists, some employed as assistants and some as privatdozents in the new Mathematical Institute. He undertook research for his doctorate with Richard Courant as his advisor and, in 1929, he was awarded the degree for his thesis

*Verallgemeinerung der Riemannschen Integrationsmethode auf Differentialgleichungen n-ter Ordnung in zwei Veränderlichen*Ⓣ. In this dissertation he generalised the Riemann's integration method, namely the explicit representation of the solution of the initial value problem of a linear hyperbolic differential equation of second order, to the case of such equations any order. Courant was able to write in Rellich's obituary with great authority about his time as a research student in Götttingen [2]:-

Of course the Nazi party destroyed the remarkable mathematical school at Göttingen but, before we look at how Rellich was affected by this, we need to explain his connection with that outstanding mathematician van der Waerden. In 1924 van der Waerden had spent seven months in Göttingen but he returned there as a visiting professor in 1929. In July of that year he met Camilla Rellich, Franz Rellich's sister, and they married in September 1929. Rellich and van der Waerden, therefore, became brothers-in-law during the time Rellich was completing his doctorate. As well as being Courant's student, Rellich was also his assistant at Göttingen, and he habilitated there in 1933. He had gone to Hamburg in the previous year where he lectured on differential and integral calculus. However, 1933 was a highly significant year for German politics since the Nazis came to power after the elections of March in that year. Rellich was a staunch opponent of National Socialism and, as a consequence, he suffered considerable problems throughout the early part of his academic career. Hermann Weyl held the chair of mathematics at Göttingen from 1930 having succeeded to Hilbert's chair. However, after the Nazis came to power, Weyl resigned his chair and accepted a position at the newly formed Institute for Advanced Study in Princeton in the United States. Segal explains the events that followed [1]:-Rellich's unique charm opened all hearts to him, his originality and his passionate interest in science made him a worthy partner[of the young mathematicians and physicists]as well as of the older members of this group. The vitality of this young generation at Göttingen seemed to justify the expectation of continuity and stability of the Institute so that Göttingen's mathematical-physical tradition was assured.

During the years 1933-34 at Göttingen, Rellich has taught courses onAfter Weyl's resignation[from Göttingen], his former assistant, Franz Rellich, became Institute Director ... Rellich had only a low-level appointment and ... was not an established figure ... There was need for a prominent mathematical figure who was suitable politically to take over the leadership in Gottingen. Furthermore, in mid-December, Rellich was ordered to report on January7for ten weeks to a field-sports camp near Berlin. This was, in fact, a mistake, since Rellich, as an Austrian citizen, was not subject to such forced training regimens. When he arrived at the camp, he was not admitted on these grounds. However, on December27, the Curator had, after some hesitation, replaced Rellich with Werner Weber as acting director of the Mathematical Institute. Rellich himself would lose his position at Gottingen six months later, on June18.

*Integral Equations and Spectral Theory*(1933) and

*Partial Differential Equations*(1934). He had also run a seminar on

*Operator Theory*in 1933-34 and one on

*Spectral Theory*in 1934. He had published significant papers such as

*Zur ersten Randwertaufgabe bei Monge-Ampèreschen Differentialgleichungen vom elliptischen Typus; differentialgeometrische Anwendungen*Ⓣ (1932),

*Über die Reduktion gewisser ausgearteter Systeme von partiellen Differentialgleichungen*Ⓣ (1934) and

*Spektraltheorie in nichtseparablen Räumen*Ⓣ (1934). Despite producing outstanding mathematics, Oswald Teichmüller and Erhard Tornier (1894-1982) considered Rellich part of the 'Courant clique' at Göttingen and, for that political (or, perhaps more accurately, racist) reason, had driven him out.

After being dismissed from Göttingen, Rellich was appointed as a privatdozent at the University of Marburg in 1934. He remained there until 1942 when he was appointed to a chair of mathematics at the University of Dresden. In September 1945 he was appointed to represent Carl Siegel's chair - Siegel had left Germany in 1940 feeling that he could no longer remain in his native land once the Nazis were in power. He worked at Princeton, accepting a permanent position there in 1946. In that year Rellich was appointed to a full Professorship to fill Siegel's chair. Courant explains the important role that Rellich played in restoring Göttingen to a world-class mathematical centre [2]:-

Rellich made many important research contributions, particularly significant being his work on perturbation theory of linear operators on a Hilbert space. Like much of Rellich's work, this was an abstract piece of mathematics motivated by questions in physics. In this particular case it was inspired by applications to quantum mechanics. Another piece of work which brought him international recognition was his study of the Monge-Ampère differential equation of elliptic type which we already mentioned when we looked at his papers published in the period 1932-34. He is also known for Rellich's theorem on entire solutions of differential equations which he proved in 1940.When he took over the leadership of the Mathematics Institute in Göttingen after its collapse, he, like no one else, was capable of building again something of the old tradition from the ashes. The dedication and selflessness with which he undertook the task is a shining example of the forces to which Germany owes its resurgence after the disaster. ... As director of the Mathematical Institute Rellich was deeply interested in creating a wide, stable base for scientific development. He succeeded in securing Göttingen's path for the foreseeable future.

A mark of the importance of his work is the fact that several of the lecture courses that he gave in the 1950s were published in the 1960s and 1970s. For example, he gave the course

*Perturbation Theory of Eigenvalue Problems*at New York University in 1953 and this was published in 1969, with a Preface by Jacob T Schwartz. Here is a quote from that Preface:-

A course which Rellich delivered at Göttingen in the winter semester of 1952-53,The present notes, taken from lectures given by Professor Rellich at New York University in1953, describe an area which he pioneered, and in which some of his most striking mathematical work was done. They show the fruitful interplay, characteristic for Rellich, of abstract operator theory with penetrating investigations of significant particular examples. In reading these notes, one comes to exceptionally close witness of the mathematical mind in the act of creation.

*Eigenwerttheorie partieller Differentialgleichungen*Ⓣ, formed a major part of the book

*Eigenwerttheorie gewöhnlicher Differentialgleichungen*Ⓣ published in 1976. The additional material in the book is written by Konrad Jörgens, who studied for his doctorate in Göttingen advised by Rellich around the time he gave the course. While we are mentioning Rellich's students, we must record that Jürgen Moser was one of Rellich's students. He received his doctorate from Göttingen in 1952 for his thesis

*Störungstheorie des kontinuierlichen Spektrums für gewöhnliche Differentialgleichungen zweiter Ordnung*Ⓣ.

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

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