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On the discrete spectrum of the Hamiltonians of \(n\)-particle systems with \(n \to \infty\) in function spaces with various permutation symmetries. (English. Russian original) Zbl 1327.81204

Funct. Anal. Appl. 49, No. 2, 148-150 (2015); translation from Funkts. Anal. Prilozh. 49, No. 2, 85-88 (2015).
Summary: The restrictions of the nonrelativistic energy operators \(H_n\) of the relative motion of a system of \(n\) identical particles with short-range interaction potentials to subspaces \(M\) of functions with various permutation symmetries are considered. It is proved that, for each of these restrictions, there exists an infinite increasing sequence of numbers \(N_j\), \(j = 1, 2, \dots\), such that the discrete spectrum of each operator \(H_{N_j}\) on \(M\) is nonempty. The family \(\{M\}\) of considered subspaces is, apparently, close to maximal among those which can be handled by the existing methods of study.

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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References:

[1] Zhislin, G. M., No article title, Teor. Mat. Fiz., 157, 116-129 (2008)
[2] L. D. Landau and E. M. Lifshits, Course of Theoretical Physics, vol. 3. Quantum Mechanics: Nonrelativistic Theory, Pergamon Press, Oxford, 2013.
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