Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 2 (2006), 024, 9 pages      math-ph/0602050

On the Essential Spectrum of Many-Particle Pseudorelativistic Hamiltonians with Permutational Symmetry Account

Grigorii Zhislin
Radiophysical Research Institute, 25/14 Bol'shaya Pechorskaya Str., Nizhny Novgorod, 603950 Russia

Received October 27, 2005, in final form February 07, 2006; Published online February 20, 2006

In this paper we formulate our results on the essential spectrum of many-particle pseudorelativistic Hamiltonians without magnetic and external potential fields in the spaces of functions, having arbitrary type α of the permutational symmetry. We discover location of the essential spectrum for all α and for some cases we establish new properties of the lower bound of this spectrum, which are useful for study of the discrete spectrum.

Key words: pseudorelativistic Hamiltonian; many-particle system; permutational symmetry; essential spectrum.

pdf (205 kb)   ps (165 kb)   tex (11 kb)


  1. Damak M., On the spectral theory of dispersive N-body Hamiltonians, J. Math. Phys., 1999, V.40, 35-48.
  2. Lewis R.T., Siedentop H., Vugalter S., The essential spectrum of relativistic multi-particle operators, Ann. Inst. H. Poincaré Phys. Théor., 1997, V.67, 1-28.
  3. Lieb E., Yau H.-T., The stability and instability of relativistic matter, Comm. Math. Phys., 1988, V.118, 177-213.
  4. Reed M., Simon B., Methods of modern mathematical physics. IV Analysis of operators, New York - San Francisco - London, Academic Press, 1978.
  5. Sigalov A.G., Sigal I.M., Invariant description, with respect to transpositions of identical particles, of the energy operator spectrum of quantum-mechanical systems, Teoret. Mat. Fiz., 1970, V.5, 73-93 (in Russian).
  6. Wigner E.P., Group theory and its application to quantum mechanics, New York, 1959.
  7. Zhislin G., Spectrum of differential operators of quantum mechanical many-particle system in the spaces of functions of the given symmetry, Izvest. Akad. Nauk SSSR, Ser. Mat., 1969, V.33, 590-649 (English transl.: Math. USSR-Izvestia, 1969, V.3, 559-616).

Previous article   Next article   Contents of Volume 2 (2006)