The nonrelativistic limit of the Dirac operator: Splitting of the operator and classification of the discrete spectrum

Thesis/Dissertation · Fri Jan 01 04:00:00 EST 1988

OSTI ID:6895508

White, G B

Two problems related to the non-relativistic limit of the Dirac operator are discussed. The first is can the Dirac operator be explicitly split into an electron and a positron part. The second is whether the discrete spectrum can be classified into electron energy levels and positron energy levels, and if so, how. Our main results are: (1)Given a relative boundedness condition on the potentials, the Dirac operator H(c) has a spectral gap of the form ({minus}mc{sup 2}+l, mc{sub 2}{minus}1) for c large enough. (2)A condition on the potential V which is sufficient to ensure that H(c) has a spectral gap at zero. (3)The standard theorems on resolvent convergence generalize to the case of pseudoresolvent convergence. (4){plus minus}H(c) {minus}mc{sup 2} converges to H{sup {plus minus}}P{sup {plus minus}} in the norm pseudoresolvent sense, where H{sup +} and H{sup {minus}} are the associated Pauli-Schroedinger operators and P{sup +} and P{sup {minus}} are the projections on the upper and lower spaces of spinors. (5)The constant {minus}l from (1) is a lower bound for H{sup +} and H{sup {minus}}.(6) Result 1 can be sharpened as follows: If {minus}K and {minus}L are lower bounds for H{sup {minus}} and H{sup +}, respectively, and {var epsilon} > 0, then H(c) has a spectral gap of the form ({minus}mc{sup 2} + K + {var epsilon}, mc{sup 2} - L - {var epsilon}) for c large enough. (7)In analogy with 4, we have ({plus minus}H(c) - mc{sup 2})Q{sup {plus minus}}(c) {yields} H{sup {plus minus}}P{sup {plus minus}} in the norm pseudoresolvent sense, where Q{sup +}(c) and Q{sup {minus}}(c) are the spectral projections of H(c) corresponding to the positive and negative halves of the spectrum respectively. (8)Using 7, the Dirac operator splits into an electron part, a positron part, and a rest mass part. (9)The existence of a spectral gap classifies the discrete spectrum into electron energy levels and positron energy levels.

• Research Organization: California Univ., Los Angeles, CA (USA)
• OSTI ID: 6895508
• Country of Publication: United States
• Language: English

Related Subjects

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71 CLASSICAL AND QUANTUM MECHANICS
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72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
99 GENERAL AND MISCELLANEOUS
990200* -- Mathematics & Computers
ANALYTIC FUNCTIONS
ANTILEPTONS
ANTIMATTER
ANTIPARTICLES
CALCULATION METHODS
DATA
DIRAC OPERATORS
ELECTRONS
ELEMENTARY PARTICLES
ENERGY LEVELS
FERMIONS
FUNCTIONS
INFORMATION
LEPTONS
MATHEMATICAL MODELS
MATHEMATICAL OPERATORS
NUMERICAL DATA
POSITRONS
QUANTUM OPERATORS
THEORETICAL DATA