Electronic Resource
College Park, Md.
:
American Institute of Physics (AIP)
Journal of Mathematical Physics
38 (1997), S. 3997-4012
ISSN:
1089-7658
Source:
AIP Digital Archive
Topics:
Mathematics
,
Physics
Notes:
We elucidate the behavior of the operator (p2+m2)1/2−α/r near the critical value α=αc where it ceases to be bounded below, by obtaining a family H(z) of operators which is self-adjoint holomorphic in a domain including all real z〉−αc−αc′, and such that H(αc−α)(α≤αc) is just the operator (p2+m2)1/2−α/r or its Friedrich extension, while H(−αc−α)(αc′〈α〈αc) is another self-adjoint extension. The operators H(z) (z real) are shown to be positive, and to have only discrete spectrum below m. The eigenvalues are then analytic functions of αc−α near α=αc (and become the eigenvalues of a non-self-adjoint operator when α〉αc). We show that these eigenvalues cannot vanish, but that the lowest eigenvalue of H(−αc−α) goes to zero when α→αc′. The L〉0 eigenvalues are analytic in α at α=αc. © 1997 American Institute of Physics.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1063/1.532106
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