Operator domains and self-adjoint operators

L. E. Ballentine, Quantum Mechanics (Prentice–Hall, Englewood Cliffs, NJ, 1990).

G.

Bonneau

,

J.

Farant

, and

G.

Valent

, “,” , – ().

T. F.

Jordan

, “,” (), – ().

A. Z.

Capri

, “,” , – ().

Vanilse S.

Araujo

,

F. A. B.

Coutinho

, and

J.

Fernando Perez

, “,” , – ().

David V. Widder, Advanced Calculus (Dover, New York, 1998) 2nd ed., p. 325.

Frigys Riesz and Bela S. Nagy, Functional Analysis (Dover, New York, 1990), 2nd ed., pp. 4–5.

E. C. Titchmarsh, The Theory of Functions (Oxford U.P., London, 1985), 2nd ed., pp. 351–353 and 366.

Thomas F. Jordan, Linear Operators for Quantum Mechanics (Wiley, New York, 1969), p. 30.

Eberhard Zeidler, Applied Functional Analysis: Applications to Mathematical Physics (Springer, New York, 1997), pp. 116–117.

We call this domain $C0∞((0,∞)\{0}).$ The symbol $C$ means that the functions are continuous. The superscript ∞ means that the functions are infinitely differentiable. The subscript 0, and the removal of the origin, \{0}, means that all the functions in the domain vanish in a small, but finite interval $[0,a]$ where $a>0$ is an arbitrary real number, and also for $x>b>a,$ another arbitrary real number.

We call this domain $C0∞(R\{0}).$ The meaning of the symbol $R\{0}$ is that the point zero has been removed from the real line. So the functions in this domain are not defined for $x=0.$ The superscript ∞ and the subscript 0 have the same meanings as before; \{0}, means that the functions $uabcd(x)$ belonging to $C0∞(R\{0})$ vanish around $x=0,$ that is for $−a<x<b$ and also for $x<−c$ and for $x>d,$ where $a,$ $b,$ $c,$ and $d$ are arbitrary positive numbers. Therefore the domain of this operator consists of functions that vanish before the point $x=0$ from the negative and positive sides and for large distances in both positive and negative directions.

F. A. B.

Coutinho

and

Y.

Nogami

, “,” , – ().

F. A. B.

Coutinho

,

Y.

Nogami

, and

J.

Fernando Perez

, “,” , – ().

F. A. B.

Coutinho

,

Y.

Nogami

, and

J.

Fernando Perez

, “,” (), – ().

M. Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1968).

J.

Fernando Perez

and

F. A. B.

Coutinho

, “,” (), – ().

S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics (Springer-Verlag, New York, 1988).

V.

Alonzo

and

S.

De Vincenzo

, “,” (), – ().

D.

Griffiths

and

S.

Walborn

, “,” (), – ().

T.

Cheon

and

T.

Shigehara

, “,” (), – ().

F. A. B.

Coutinho

, “,” (), – ().

M. Reed and B. Simon, Methods of Modern Mathematical Physics (Academic, London, 1975), Vol. 2.

In the mathematical literature the two equations (A5) and (A6) are written with O replaced by $O+.$ In this way we don’t need to say anything about the domain.