Self-Adjoint Operators, the Friedrichs Extension, and the Partial Differential Equations of Mathematical Physics

Abstract

In this Chapter we want to study the following problems in a Hilbert space X, wehere A: D(A) ⊆ X → X is a linear symmetric operator that has additional properties to be discussed ahead (cf. also Figure 5.1).

In the fall of 1926, the young John von Neumann (1903–1957) arrived at Göttingen to take up his duties as Hilbert’s assistant. These were the hectic years during which quantum mechanics was developing at breakneck speed, with a new idea popping up every few weeks from all over the horizon. Jean Dieudonné History of Functional Analysis, 1981

Stimulated by an interest in quantum mechanics, John von Neumann (1903–1957) began the work in operator theory. The result was a paper von Neumann submitted for publication to the Mathematische Zeitschrift but later withdrew. The reason for this withdrawal was that in 1928 Erhard Schmidt and myself, independently, saw the role which could be played in the theory by the concept of the adjoint operator, and the importance which should be attached to self-adjoint operators. When von Neumann learned from Professor Schmidt of this observation, he was able to rewrite his paper in a much more satisfactory and complete form. Incidentally, for permission to withdraw the paper, the publisher exacted from Professor von Neumann a promise to write a book on quantum mechanics. The book soon appeared and has become one of the classics of modern physics (Foundations of Quantum Mechanics, Springer-Verlag, 1932). Marshall Harvey Stone, 1970

At the very beginning the given elliptic differential operator is only defined on a space of C²-functions. This operator is extended to an abstractly defined operator by using a formal closure. The main task is to show that the extended operator is self-adjoint. In this case it is possible to apply the methods of John von Neumann. Kurt Otto Friedrichs, 1934

The fundamental quality required of operators representing physical quantities in quantum mechanics is that they be self-adjoint which is equivalent to saying that the eigenvalue problem is completely solvable for them, that is, there exists a complete set (discrete or continuous) of eigenfunctions.

The problem has, of course, been solved in the case of operators for which the eigenvalue problem is explicitly solved by separation of variables or other methods, but it seems not have been settled in the general case of many-particle systems.

The main purpose of the present paper is to show that the Schrödinger Hamiltonian operator of every atom, molecule, or ion, in short, of every system composed of a finite number of particles interacting with each other through a potential energy, for instance, of Coulomb type, is essentially self-adjoint (i.e., this operator possesses a unique self-adjoint extension). Thus, our result serves as a mathematical basis for all theoretical works concerning nonrelativistic quantum mechanics. Tosio Kato, 1951

The interaction between physics and mathematics has always played an important role. The physicist who does not have the latest mathematical knowledge available to him is at a distinct disadvantage. The mathematician who shies away from physical applications will most likely miss important insights and motivations. Martin Schechter, 1981