Glasnik Matematicki, Vol. 40, No.2 (2005), 261-301.

S-DIAGONALIZABLE OPERATORS IN QUANTUM MECHANICS

David Carfi

Abstract.   In this paper we study a certain class of endomorphisms on the space of tempered distributions. More precisely, the core of the paper deals with endomorphisms, defined on the whole space of tempered distributions, for which there exists an S-basis of the space formed by their eigenvectors. We call these operators S-diagonalizable operators. One of the goals of the paper is the realization that this class of endomorphisms represents in the infinite-dimensional case what in finite-dimension is represented by the diagonalizable matrices.

We concentrate our examination on two aspects: the study of the spectrum of these operators and the foundation of a functional calculus for them. Concerning the first aspect, we do not assume nothing about the spectrum of these operators. The circumstance that the eigenvalue-spectrum of these operators will be continuous (more precisely it will be a connected subset of the complex plane, as it is proved in the present note) is a consequence of our definition. Moreover, the spectral measures will be not used in the construction of the functional calculus. In such a way, the definition of the function of an operator, presented in the paper, differs deeply from the usual one, in which the spectral measures of the operators play a fundamental role (as in the spectral decomposition of an operator). Note that, even in the case in which the eigenvalues-spectrum is a subset of the real line, we show that it is not necessarily coinciding with the whole real line.

2000 Mathematics Subject Classification.   46F10, 46F99, 47A05, 47N50, 70A05, 70B05, 81P05, 81Q99.

Key words and phrases.   Linear operator, tempered distribution, basis, linear superposition, eigenvalue, diagonalizability.

DOI: 10.3336/gm.40.2.08

References:

1. J. Barros-Neto, An Introduction to the theory of distributions, Marcel Dekker, Inc. NewYork, 1973.

2. N. Boccara, Functional analysis, an introduction for physicists, Academic press, Inc. 1990.

3. A. Bohm and M. Gadella, Dirac kets, Gamow vectors and Gel'fand triplets. The rigged Hilbert space formulation of quantuum mechanics, Lecture Notes in Physics 348, Springer-Verlag, Berlin, 1989.

4. D. Carfì, S-linear operators in quantum mechanics and in economy, Appl. Sci. 6 (2004) 7-20.

5. D. Carfì, Dirac-orthogonality in the space of tempered distributions, J. Comput. Appl. Math. 153 (2003), 99-107.
   CrossRef

6. P. A. M. Dirac, The principles of Quantum Mechanics, Oxford Claredon press, 1930.

7. G. Gilardi, Analisi 3, McGraw-Hill, 1994.

8. J. Horvath, Topological Vector Spaces and Distributions (Vol.I), Addison-Wesley Publishing Company, 1966.

9. R. Shankar, Principles of Quantum Mechanics, Plenum Press, 1994.