Advertisement

The Discrete Spectra of the Dirac and Pauli Operators

  • O. I. Kurbenin
Part of the Topics in Mathematical Physics book series (TOMP, volume 3)

Abstract

The aim of the present article is the investigation of the spectra of the Dirac and Pauli operators by operator-theoretic methods. This investigation is based on the estimation of the quadratic forms of the above operators by means of the quadratic forms of operators with well-known spectra. We may count the Schroedinger operator among the latter. Estimates of this type extended to both sides allow us, in some cases, to establish the criteria for the total multiplicity of the spectrum of the perturbed operator to be finite or infinite in that part of the axis which min the case of the unperturbed operator is a gap (i.e., free from the spectrum). Qualitative conclusions on the character of the spectrum and quantitative estimates of the total multiplicity of the spectrum follow from the well-known properties of the Schroedinger operator. In some cases, these conclusions refer to a family of operators obtained through the introduction of a parameter h which plays the part of Planck’s constant in quantum mechanics. Skachek [6] has recently obtained the lower bound to the number of eigenvalues of the Dirac operator. By contrast with the results of [6], our estimates do not contain terms involving the derivative of the potential. Some results concerning the spectrum of the Pauli operator have been obtained by Glazman [4].

Keywords

Dirac Operator Discrete Spectrum Linear Manifold Pauli Operator Finite Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literature Cited

  1. 1.
    M. Sh. Birman, “The spectra of singular boundary problems,” Matem. Sbornik, Vol. 55 (97) (1961).Google Scholar
  2. 2.
    V. A. Fok, The Origins of Quantum Mechanics, Izd. LGU (1932).Google Scholar
  3. 3.
    H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One and Two Electron Atoms Academic Press, New York (1957).Google Scholar
  4. 4.
    I. M. Glazman, Direct Methods for the Spectral Analysis of Singular Differential Operators, Izd. Nauka (1963).Google Scholar
  5. 5.
    L.D. Landau and E. M. Lifshits, Quantum Mechanics ( Nonrelativistic Theory, Fizmatgiz (1963).Google Scholar
  6. 6.
    B. Ya. Skachek, “A remark concerning the spectrum of the Dirac operator,” Dokl. Akad. Nauk Ukrain.SSR, Vol. 65 (10) (1965).Google Scholar

Copyright information

© Consultants Bureau, New York 1969

Authors and Affiliations

  • O. I. Kurbenin

There are no affiliations available

Personalised recommendations